L(s) = 1 | + (−1.73 − i)2-s + (−0.202 + 0.351i)3-s + (1.99 + 3.46i)4-s − 6.36i·5-s + (0.702 − 0.405i)6-s + (−2.20 + 1.27i)7-s − 7.99i·8-s + (13.4 + 23.2i)9-s + (−6.36 + 11.0i)10-s + (−22.6 − 13.0i)11-s − 1.62·12-s + 5.10·14-s + (2.23 + 1.29i)15-s + (−8 + 13.8i)16-s + (−46.8 − 81.1i)17-s − 53.6i·18-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.0390 + 0.0676i)3-s + (0.249 + 0.433i)4-s − 0.569i·5-s + (0.0478 − 0.0276i)6-s + (−0.119 + 0.0688i)7-s − 0.353i·8-s + (0.496 + 0.860i)9-s + (−0.201 + 0.348i)10-s + (−0.620 − 0.357i)11-s − 0.0390·12-s + 0.0973·14-s + (0.0384 + 0.0222i)15-s + (−0.125 + 0.216i)16-s + (−0.668 − 1.15i)17-s − 0.702i·18-s + ⋯ |
Λ(s)=(=(338s/2ΓC(s)L(s)(−0.978−0.207i)Λ(4−s)
Λ(s)=(=(338s/2ΓC(s+3/2)L(s)(−0.978−0.207i)Λ(1−s)
Degree: |
2 |
Conductor: |
338
= 2⋅132
|
Sign: |
−0.978−0.207i
|
Analytic conductor: |
19.9426 |
Root analytic conductor: |
4.46571 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ338(147,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 338, ( :3/2), −0.978−0.207i)
|
Particular Values
L(2) |
≈ |
0.1719298671 |
L(21) |
≈ |
0.1719298671 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.73+i)T |
| 13 | 1 |
good | 3 | 1+(0.202−0.351i)T+(−13.5−23.3i)T2 |
| 5 | 1+6.36iT−125T2 |
| 7 | 1+(2.20−1.27i)T+(171.5−297.i)T2 |
| 11 | 1+(22.6+13.0i)T+(665.5+1.15e3i)T2 |
| 17 | 1+(46.8+81.1i)T+(−2.45e3+4.25e3i)T2 |
| 19 | 1+(−32.2+18.6i)T+(3.42e3−5.94e3i)T2 |
| 23 | 1+(52.4−90.8i)T+(−6.08e3−1.05e4i)T2 |
| 29 | 1+(124.−216.i)T+(−1.21e4−2.11e4i)T2 |
| 31 | 1+278.iT−2.97e4T2 |
| 37 | 1+(−9.43−5.45i)T+(2.53e4+4.38e4i)T2 |
| 41 | 1+(321.+185.i)T+(3.44e4+5.96e4i)T2 |
| 43 | 1+(206.+358.i)T+(−3.97e4+6.88e4i)T2 |
| 47 | 1−238.iT−1.03e5T2 |
| 53 | 1+424.T+1.48e5T2 |
| 59 | 1+(670.−387.i)T+(1.02e5−1.77e5i)T2 |
| 61 | 1+(−61.7−106.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(763.+440.i)T+(1.50e5+2.60e5i)T2 |
| 71 | 1+(102.−59.3i)T+(1.78e5−3.09e5i)T2 |
| 73 | 1−209.iT−3.89e5T2 |
| 79 | 1+532.T+4.93e5T2 |
| 83 | 1−376.iT−5.71e5T2 |
| 89 | 1+(36.9+21.3i)T+(3.52e5+6.10e5i)T2 |
| 97 | 1+(553.−319.i)T+(4.56e5−7.90e5i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.65806177800329369280697919977, −9.620091094079445647982478740671, −8.906077624853091578543913948776, −7.83457770431815100711643462173, −7.09479644484032353683100621637, −5.52276190185409159991818365410, −4.57422993562439570415399552299, −3.03000248180092872780894719604, −1.68115457801384361916567776499, −0.07303609752524847649026298262,
1.69789291561701999109720965013, 3.25961829442592949411396701286, 4.67998326115664461177905174740, 6.19667191462973407556084609596, 6.76128620214845374942417424214, 7.82793336704261732622310828214, 8.750270784937242225844225049007, 9.899394494387568659052652750907, 10.40168979664307558985193189943, 11.43429924126071776454059155976