I have a rooted .290 however mine is unlocked bootloader lah. I'll teach the LB method below.
Not necessary. Just use XZ Dual Recovery.
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Before you download anything or start with updating to 4.3 and root, read through the guide fully first as STEP 4 has two options. I'd say go with option 2 as less hassle - less 1 file (and few hundreds Mb) to download.

Also, I recommend to download nandroid from [url=https://play.google.com/store/apps/details?id=com.h3r3t1c.onnandbup&hl=en]PLAY STORE[/url] while you are still on .257 rooted. Back up entirely, to SDCARD1.

And to back up your TRIM AREA (TA) [url=http://forum.xda-developers.com/showthread.php?t=2292598]GUIDE HERE[/url]

If your computer, device blows up, burns your house down, The Ring ghost comes out of your TV, I am not responsible.

Move on if you agree to the terms & conditions.

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[center][color=blue][size=7]How to root .290[/size][/color][/center]

I'll go with step by step, instead of skipping .257 update as I did it this way but I'll post a non-root method here. Mine's through unlocked BL. Anything below marked with asterisk (*) means locked bootloader method, marked with hash (#) means unlocked bootloader. Not marked means both ways.
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Before we begin, files you will need
a) [url=http://www.flashtool.net/download.php]Flashtool - download the latest one for Z1 support![/url]
b) [url=http://developer.android.com/sdk/index.html#download]Android SDK (a must)[/url]
c) [not entirely necessary] [url=https://mega.co.nz/#!nwoGmKYA!GT4X-i4LD2ySJcBPORHFlCZ2R_l9Y3O000PpcThstXo].257 FTF firmware[/url]
d) [url=http://forum.xda-developers.com/showthread.php?t=1886460]Root tool by Bin4ry[/url]
e) # [url=https://mega.co.nz/#!39oRCSpI!MjlOT2T5TIl47kuRIhXKwgmS9XBZyXQKaKgnvV1Iu3c].290 FTF firmware[/url]
f) * [url=http://forum.xda-developers.com/showthread.php?t=2572988]XZ Dual Recovery & .290 flashable ZIP[/url]
g) * [url=http://forum.xda-developers.com/showthread.php?t=2572988]Flashable .290 FTF[/url] & a [url=http://nut.xperia-files.com/?dlid=QzY5MDNfMTQuMi5BLjAuMjkwLnBhdGNo]MUST PATCH FILE.zip[/url]
h) * [url=http://nut.xperia-files.com/?dlid=QzY5MDNfMTQuMi5BLjAuMjkwLnBhdGNo]Stripped FTF for FLASHTOOL to flash[/url]
i) at least 4Gb SDcard
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Pre-rooting to-do list.

1) Download all the files required and provided above - also to put them in separate folders named according to steps for your convenience, if you like.

2) Open SDKManager.exe that you downloaded from Android SDK.zip
Install the following :
a) TOOLS\Android SDK Platform-tools
b) Android 4.2.2 (API 17)\SDK Platform; Google APIs; Sources for Android SDK
c) EXTRAS\Android Support Library & Google USB driver

3) Place downloaded .FTF firmware files in the FLASHTOOL\firmwares folder.
4) Place downloaded .zip files in their supposedly locations, which is your SDCard.

Important note : SDCard on your phone will be shown as SDCARD1 , SDCARD0 is your phone internal memory.
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Step 1 -
Are you already on a rooted .257? Yes - proceed to Step 4; No - go to step 2. For Unlocked Bootloader method, see STEP 7 onwards.
Step 2 -
.257 does not officially have any root yet, hence you need to flash back to .534 (entirely, including wiping data, to prevent data conflict, bootloops.)
*Step 3 - FLASHING .534 ftf
> Done with flashing .534; boot into normal screen , use Bin4ry's root tool provided above.
> On this tool, you must enable "USB Debugging" and enable "Unknown sources" in Settings > Security.

===how to enable USB debugging===
Go to Settings > About > Tap BUILD NUMBER constantly for 10 times or so. You will get "YOU ARE NOW A DEVELOPER". Back out, you should see "Developer options". Go inside and enable USB debugging/

> On this tool, you can select option 0, 1 or 2, up to you. They're all the same but I used option 0.
> Rooting .534 is done.
*Step 4 OPTION 1 - you are still .534 here.
> Open Flashtool.
> Click on LIGHTNING icon on Flashtool
> Click Flashmode
> Select .257 firmware
[color=red]!!!IMPORTANT!!!
REMOVE THE [b][i]TICK[/i][/b] ON [i]WIPE DATA AND CACHE[/i][/color]
> Get your phone ready in flash mode by switching off, then VOLUME DOWN + connect USB cable. You should hear some audio on your PC that a device is connected.
> Start flash
*Step 4 OPTION 2 - Still on .534
> Update to .257 from OTA.
> Done.
You still keep your root.
*Step 5 - Now you should be on .257 with root. Double check by opening SuperSU and see if you have any errors.
> If no errors, proceed.
> Install XZ Dual Recovery via Windows, follow the guide [url=http://forum.xda-developers.com/showthread.php?t=2261606]HERE[/url] to install DUAL RECOVERY.
> If install is successful, try to boot into recovery - to try.
> Quickly

To boot into recovery, keep pressing VOLUME UP or DOWN.
Up = CWM
Down = TWRP

*Step 6 -
> Copy FLASHABLE .290 FTF to your SDcard.
> Boot into recovery again
> Flash .290.zip in recovery, use TWRP to ease next step.
> REBOOT TO RECOVERY AGAIN
> You have 40 seconds to do this.
> [i]Flash the "MUST FLASH THIS.zip" I provided above to fix the 40-seconds-recovery-reboot. [/i]
> Select REBOOT in TWRP but under reboot section, select "POWER OFF"
> VOLUME DOWN and connect USB for flashmode.
> Flash STRIPPED FTF via Flashtool. Do not remove anything, let it be default.
[size=7]DONE.[/size]
Reboot. You should be all good to go now, first boot may take longer time.
[center][color=green][size=7]# UNLOCKED BOOTLOADER method[/size]
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Basically starting from STEP 5.

Now you have .257 rooted, locked bootloader. Remember to BACK UP TRIM AREA first before you proceed to unlock BL and a full back up using Nandroid.

Unlock your bootloader first, you can do so using Flashtool by clicking "BLU" next to the LIGHTNING icon or you can do the geek method - which you will mindfark if you're not familiar with command prompt, so I guess I won't post the 2nd method.

Now that your bootloader is unlocked.

Firstly :
--Download [url=http://download.chainfire.eu/372/SuperSU/UPDATE-SuperSU-v1.86.zip]SuperSU[/url] and place in your SDcard1.
> Flash .290 FTF via Flashtool but be very careful on this part.
A) Remove the check on "WIPE DATA" "WIPE CACHE"
B) FLASH. Let it complete and start up the device.
> Confirm on your device that you STILL HAVE SuperSU. But it won't be working, you'll get some binary error.

> Flash DooMKernel from [url=http://forum.xda-developers.com/showthread.php?t=2448613]GUIDE HERE[/url]
> Boot into TWRP/CWM, recommended using TWRP as you have the option to POWER OFF.
> Navigate and flash SuperSU.zip
> Reboot.

[size=7]DONE.[/size]
More guides...
IMEI is now 0000000000000 ?
i) flash the same country fw with your current .290
ii) flash commercial / journalist .534
[credit to ZeneticX for giving precise details, and original credits to its real founder who found the fix.]