L(s) = 1 | + (1.21 + 0.722i)2-s + (0.5 + 0.866i)3-s + (0.957 + 1.75i)4-s + (−0.403 + 0.403i)5-s + (−0.0173 + 1.41i)6-s + (4.65 − 1.24i)7-s + (−0.103 + 2.82i)8-s + (−0.499 + 0.866i)9-s + (−0.782 + 0.199i)10-s + (−5.72 − 1.53i)11-s + (−1.04 + 1.70i)12-s + (−0.849 − 3.50i)13-s + (6.55 + 1.84i)14-s + (−0.551 − 0.147i)15-s + (−2.16 + 3.36i)16-s + (−0.917 − 0.529i)17-s + ⋯ |
L(s) = 1 | + (0.859 + 0.510i)2-s + (0.288 + 0.499i)3-s + (0.478 + 0.878i)4-s + (−0.180 + 0.180i)5-s + (−0.00707 + 0.577i)6-s + (1.75 − 0.471i)7-s + (−0.0367 + 0.999i)8-s + (−0.166 + 0.288i)9-s + (−0.247 + 0.0630i)10-s + (−1.72 − 0.462i)11-s + (−0.300 + 0.492i)12-s + (−0.235 − 0.971i)13-s + (1.75 + 0.492i)14-s + (−0.142 − 0.0381i)15-s + (−0.541 + 0.840i)16-s + (−0.222 − 0.128i)17-s + ⋯ |
Λ(s)=(=(312s/2ΓC(s)L(s)(0.280−0.959i)Λ(2−s)
Λ(s)=(=(312s/2ΓC(s+1/2)L(s)(0.280−0.959i)Λ(1−s)
Degree: |
2 |
Conductor: |
312
= 23⋅3⋅13
|
Sign: |
0.280−0.959i
|
Analytic conductor: |
2.49133 |
Root analytic conductor: |
1.57839 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ312(115,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 312, ( :1/2), 0.280−0.959i)
|
Particular Values
L(1) |
≈ |
1.87433+1.40572i |
L(21) |
≈ |
1.87433+1.40572i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.21−0.722i)T |
| 3 | 1+(−0.5−0.866i)T |
| 13 | 1+(0.849+3.50i)T |
good | 5 | 1+(0.403−0.403i)T−5iT2 |
| 7 | 1+(−4.65+1.24i)T+(6.06−3.5i)T2 |
| 11 | 1+(5.72+1.53i)T+(9.52+5.5i)T2 |
| 17 | 1+(0.917+0.529i)T+(8.5+14.7i)T2 |
| 19 | 1+(−0.617+0.165i)T+(16.4−9.5i)T2 |
| 23 | 1+(−1.02−1.77i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−4.86+2.81i)T+(14.5−25.1i)T2 |
| 31 | 1+(2.54−2.54i)T−31iT2 |
| 37 | 1+(−1.83+6.84i)T+(−32.0−18.5i)T2 |
| 41 | 1+(−1.37+5.12i)T+(−35.5−20.5i)T2 |
| 43 | 1+(8.58+4.95i)T+(21.5+37.2i)T2 |
| 47 | 1+(5.92+5.92i)T+47iT2 |
| 53 | 1+1.19iT−53T2 |
| 59 | 1+(−2.11−7.89i)T+(−51.0+29.5i)T2 |
| 61 | 1+(−5.30−3.06i)T+(30.5+52.8i)T2 |
| 67 | 1+(2.06−7.71i)T+(−58.0−33.5i)T2 |
| 71 | 1+(−0.113−0.422i)T+(−61.4+35.5i)T2 |
| 73 | 1+(−7.34+7.34i)T−73iT2 |
| 79 | 1−7.20iT−79T2 |
| 83 | 1+(−7.79−7.79i)T+83iT2 |
| 89 | 1+(0.708+0.189i)T+(77.0+44.5i)T2 |
| 97 | 1+(4.48−1.20i)T+(84.0−48.5i)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
−11.80126851390103744018520781765, −10.94971158950878722184143677308, −10.41663550753948944603668816051, −8.572224232052077672045614458213, −7.914859789168105482097744261705, −7.28819024887384992833941975143, −5.33295879487859266206439775568, −5.11353850305542926006385934902, −3.73829158164811950369885511371, −2.46706692532340034561219214500,
1.72955028165285519362255284505, 2.69148212257369118143788401698, 4.62272924140806578521424537066, 5.02920980458220544177771144573, 6.46959502542741543742210119301, 7.73839109770677943779067740037, 8.417268912617238827666829132441, 9.827435878120612301959142816887, 10.92506088137151534969379597337, 11.60724817434740655909225516300