L(s) = 1 | + (0.5 + 0.866i)2-s + (1.98 + 0.531i)3-s + (−0.499 + 0.866i)4-s + (0.531 + 1.98i)6-s + (2.67 + 1.54i)7-s − 0.999·8-s + (1.05 + 0.609i)9-s + (−0.858 + 3.20i)11-s + (−1.45 + 1.45i)12-s + (−1.43 − 3.30i)13-s + 3.09i·14-s + (−0.5 − 0.866i)16-s + (1.44 + 5.40i)17-s + 1.21i·18-s + (0.296 − 0.0794i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (1.14 + 0.306i)3-s + (−0.249 + 0.433i)4-s + (0.217 + 0.809i)6-s + (1.01 + 0.584i)7-s − 0.353·8-s + (0.351 + 0.203i)9-s + (−0.258 + 0.965i)11-s + (−0.419 + 0.419i)12-s + (−0.397 − 0.917i)13-s + 0.826i·14-s + (−0.125 − 0.216i)16-s + (0.351 + 1.31i)17-s + 0.287i·18-s + (0.0680 − 0.0182i)19-s + ⋯ |
Λ(s)=(=(650s/2ΓC(s)L(s)(0.00940−0.999i)Λ(2−s)
Λ(s)=(=(650s/2ΓC(s+1/2)L(s)(0.00940−0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
650
= 2⋅52⋅13
|
Sign: |
0.00940−0.999i
|
Analytic conductor: |
5.19027 |
Root analytic conductor: |
2.27821 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ650(457,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 650, ( :1/2), 0.00940−0.999i)
|
Particular Values
L(1) |
≈ |
1.86359+1.84614i |
L(21) |
≈ |
1.86359+1.84614i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5−0.866i)T |
| 5 | 1 |
| 13 | 1+(1.43+3.30i)T |
good | 3 | 1+(−1.98−0.531i)T+(2.59+1.5i)T2 |
| 7 | 1+(−2.67−1.54i)T+(3.5+6.06i)T2 |
| 11 | 1+(0.858−3.20i)T+(−9.52−5.5i)T2 |
| 17 | 1+(−1.44−5.40i)T+(−14.7+8.5i)T2 |
| 19 | 1+(−0.296+0.0794i)T+(16.4−9.5i)T2 |
| 23 | 1+(−1.35+5.05i)T+(−19.9−11.5i)T2 |
| 29 | 1+(−6.38+3.68i)T+(14.5−25.1i)T2 |
| 31 | 1+(1.42−1.42i)T−31iT2 |
| 37 | 1+(3.19−1.84i)T+(18.5−32.0i)T2 |
| 41 | 1+(−9.05−2.42i)T+(35.5+20.5i)T2 |
| 43 | 1+(9.26−2.48i)T+(37.2−21.5i)T2 |
| 47 | 1+5.28iT−47T2 |
| 53 | 1+(−2.01+2.01i)T−53iT2 |
| 59 | 1+(0.224+0.837i)T+(−51.0+29.5i)T2 |
| 61 | 1+(−2.36+4.09i)T+(−30.5−52.8i)T2 |
| 67 | 1+(1.62+2.81i)T+(−33.5+58.0i)T2 |
| 71 | 1+(1.79+6.68i)T+(−61.4+35.5i)T2 |
| 73 | 1+11.2T+73T2 |
| 79 | 1−4.04iT−79T2 |
| 83 | 1+5.45iT−83T2 |
| 89 | 1+(−7.03−1.88i)T+(77.0+44.5i)T2 |
| 97 | 1+(−7.07+12.2i)T+(−48.5−84.0i)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
−10.51597252886191540978257088736, −9.796365189256632630731523939119, −8.623870253531542829537106237408, −8.251242790229907383868376182160, −7.53601899801849326695923745279, −6.26190842023048159718498064738, −5.14356388931791761584564980983, −4.36813181251116186088638028676, −3.13191715693818202241111398177, −2.09613014773916742796736927559,
1.30040169316445446672658488530, 2.53770619919311891945121610259, 3.44379358313239969175541892278, 4.59702000700249379146255453955, 5.52702060573578745499643153113, 7.08309095629580402890644412438, 7.76015632624260345140725978060, 8.718011345520474719819010659241, 9.335297228745494053481769967773, 10.42197103525457351139410786702