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Properties

Label 2-637-1.1-c1-0-4
Degree 22
Conductor 637637
Sign 11
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.656·2-s − 0.204·3-s − 1.56·4-s − 1.35·5-s + 0.134·6-s + 2.34·8-s − 2.95·9-s + 0.892·10-s − 1.90·11-s + 0.321·12-s + 13-s + 0.278·15-s + 1.60·16-s + 3.56·17-s + 1.94·18-s + 0.985·19-s + 2.13·20-s + 1.25·22-s + 1.69·23-s − 0.479·24-s − 3.15·25-s − 0.656·26-s + 1.21·27-s + 6.54·29-s − 0.182·30-s + 7.69·31-s − 5.73·32-s + ⋯
L(s)  = 1  − 0.463·2-s − 0.118·3-s − 0.784·4-s − 0.608·5-s + 0.0548·6-s + 0.828·8-s − 0.986·9-s + 0.282·10-s − 0.574·11-s + 0.0927·12-s + 0.277·13-s + 0.0718·15-s + 0.400·16-s + 0.864·17-s + 0.457·18-s + 0.226·19-s + 0.477·20-s + 0.266·22-s + 0.353·23-s − 0.0978·24-s − 0.630·25-s − 0.128·26-s + 0.234·27-s + 1.21·29-s − 0.0333·30-s + 1.38·31-s − 1.01·32-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 11
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 1)(2,\ 637,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.68667955230.6866795523
L(12)L(\frac12) ≈\approx 0.68667955230.6866795523
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
13 1−T 1 - T
good2 1+0.656T+2T2 1 + 0.656T + 2T^{2}
3 1+0.204T+3T2 1 + 0.204T + 3T^{2}
5 1+1.35T+5T2 1 + 1.35T + 5T^{2}
11 1+1.90T+11T2 1 + 1.90T + 11T^{2}
17 1−3.56T+17T2 1 - 3.56T + 17T^{2}
19 1−0.985T+19T2 1 - 0.985T + 19T^{2}
23 1−1.69T+23T2 1 - 1.69T + 23T^{2}
29 1−6.54T+29T2 1 - 6.54T + 29T^{2}
31 1−7.69T+31T2 1 - 7.69T + 31T^{2}
37 1+2.02T+37T2 1 + 2.02T + 37T^{2}
41 1−9.88T+41T2 1 - 9.88T + 41T^{2}
43 1+3.16T+43T2 1 + 3.16T + 43T^{2}
47 1−7.76T+47T2 1 - 7.76T + 47T^{2}
53 1−0.354T+53T2 1 - 0.354T + 53T^{2}
59 1−2.16T+59T2 1 - 2.16T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1−11.3T+67T2 1 - 11.3T + 67T^{2}
71 1+9.05T+71T2 1 + 9.05T + 71T^{2}
73 1+7.13T+73T2 1 + 7.13T + 73T^{2}
79 1+5.39T+79T2 1 + 5.39T + 79T^{2}
83 1+2.03T+83T2 1 + 2.03T + 83T^{2}
89 1+6.89T+89T2 1 + 6.89T + 89T^{2}
97 1+14.6T+97T2 1 + 14.6T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.43370309948356759355377065584, −9.745838140221454045526974486032, −8.633099350450562745822207012641, −8.193375193607149075719694659386, −7.34250280098881812049235129319, −5.94653348036128069079180912338, −5.07536770607284294125650802195, −4.01975644818778433237683685396, −2.84860105763173388855996623049, −0.77488385266128951688398560546, 0.77488385266128951688398560546, 2.84860105763173388855996623049, 4.01975644818778433237683685396, 5.07536770607284294125650802195, 5.94653348036128069079180912338, 7.34250280098881812049235129319, 8.193375193607149075719694659386, 8.633099350450562745822207012641, 9.745838140221454045526974486032, 10.43370309948356759355377065584

Graph of the ZZ-function along the critical line