L(s) = 1 | + (0.588 − 1.01i)2-s + (−1.67 − 2.90i)3-s + (0.308 + 0.533i)4-s + (−1.57 + 2.72i)5-s − 3.94·6-s + 3.07·8-s + (−4.11 + 7.12i)9-s + (1.85 + 3.20i)10-s + (0.386 + 0.669i)11-s + (1.03 − 1.78i)12-s + 13-s + 10.5·15-s + (1.19 − 2.06i)16-s + (2.87 + 4.98i)17-s + (4.83 + 8.38i)18-s + (0.611 − 1.05i)19-s + ⋯ |
L(s) = 1 | + (0.415 − 0.720i)2-s + (−0.967 − 1.67i)3-s + (0.154 + 0.266i)4-s + (−0.704 + 1.21i)5-s − 1.60·6-s + 1.08·8-s + (−1.37 + 2.37i)9-s + (0.585 + 1.01i)10-s + (0.116 + 0.201i)11-s + (0.298 − 0.516i)12-s + 0.277·13-s + 2.72·15-s + (0.298 − 0.516i)16-s + (0.697 + 1.20i)17-s + (1.14 + 1.97i)18-s + (0.140 − 0.242i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.947+0.318i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.947+0.318i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.947+0.318i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.947+0.318i)
|
Particular Values
L(1) |
≈ |
1.18869−0.194406i |
L(21) |
≈ |
1.18869−0.194406i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1−T |
good | 2 | 1+(−0.588+1.01i)T+(−1−1.73i)T2 |
| 3 | 1+(1.67+2.90i)T+(−1.5+2.59i)T2 |
| 5 | 1+(1.57−2.72i)T+(−2.5−4.33i)T2 |
| 11 | 1+(−0.386−0.669i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−2.87−4.98i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−0.611+1.05i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−1.49+2.59i)T+(−11.5−19.9i)T2 |
| 29 | 1+2.46T+29T2 |
| 31 | 1+(−3.06−5.31i)T+(−15.5+26.8i)T2 |
| 37 | 1+(2.49−4.32i)T+(−18.5−32.0i)T2 |
| 41 | 1+2.55T+41T2 |
| 43 | 1+2.73T+43T2 |
| 47 | 1+(2.68−4.65i)T+(−23.5−40.7i)T2 |
| 53 | 1+(−4.89−8.47i)T+(−26.5+45.8i)T2 |
| 59 | 1+(1.25+2.16i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−5.45+9.45i)T+(−30.5−52.8i)T2 |
| 67 | 1+(2.16+3.74i)T+(−33.5+58.0i)T2 |
| 71 | 1−10.6T+71T2 |
| 73 | 1+(−2.58−4.48i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−0.271+0.469i)T+(−39.5−68.4i)T2 |
| 83 | 1−15.2T+83T2 |
| 89 | 1+(4.61−7.99i)T+(−44.5−77.0i)T2 |
| 97 | 1+1.26T+97T2 |
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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
−10.90498238355645624648308928172, −10.38592953046474334582590630908, −8.258895861851563418286777164625, −7.71881962462945203706729332311, −6.85146845471497203510902445510, −6.40514107658929293048226730825, −5.06092016791022582861657263712, −3.58802049802836212621727640326, −2.58697800231998390392696015595, −1.41336570396533091983992863286,
0.71832641340275595539265413317, 3.61606488902854498404339955160, 4.41297229559688395789238665096, 5.26596210631090284479308473197, 5.56558715453895577651576444654, 6.78621408059178065761648644496, 8.023343557651541048360356550510, 9.076713134543225759603710745685, 9.775845539081196593750151760289, 10.59687253695894961458499949695