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Properties

Label 2-637-1.1-c1-0-33
Degree 22
Conductor 637637
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.86·2-s − 3.34·3-s + 1.48·4-s + 0.866·5-s − 6.24·6-s − 0.965·8-s + 8.21·9-s + 1.61·10-s − 3.86·11-s − 4.96·12-s + 13-s − 2.90·15-s − 4.76·16-s − 3.34·17-s + 15.3·18-s − 5.38·19-s + 1.28·20-s − 7.21·22-s − 5.24·23-s + 3.23·24-s − 4.24·25-s + 1.86·26-s − 17.4·27-s + 1.69·29-s − 5.41·30-s + 7.56·31-s − 6.96·32-s + ⋯
L(s)  = 1  + 1.31·2-s − 1.93·3-s + 0.741·4-s + 0.387·5-s − 2.55·6-s − 0.341·8-s + 2.73·9-s + 0.511·10-s − 1.16·11-s − 1.43·12-s + 0.277·13-s − 0.748·15-s − 1.19·16-s − 0.812·17-s + 3.61·18-s − 1.23·19-s + 0.287·20-s − 1.53·22-s − 1.09·23-s + 0.659·24-s − 0.849·25-s + 0.365·26-s − 3.36·27-s + 0.315·29-s − 0.988·30-s + 1.35·31-s − 1.23·32-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: −1-1
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: 11
Selberg data: (2, 637, ( :1/2), −1)(2,\ 637,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−1.86T+2T2 1 - 1.86T + 2T^{2}
3 1+3.34T+3T2 1 + 3.34T + 3T^{2}
5 1−0.866T+5T2 1 - 0.866T + 5T^{2}
11 1+3.86T+11T2 1 + 3.86T + 11T^{2}
17 1+3.34T+17T2 1 + 3.34T + 17T^{2}
19 1+5.38T+19T2 1 + 5.38T + 19T^{2}
23 1+5.24T+23T2 1 + 5.24T + 23T^{2}
29 1−1.69T+29T2 1 - 1.69T + 29T^{2}
31 1−7.56T+31T2 1 - 7.56T + 31T^{2}
37 1+4.83T+37T2 1 + 4.83T + 37T^{2}
41 1+4.06T+41T2 1 + 4.06T + 41T^{2}
43 1−4.03T+43T2 1 - 4.03T + 43T^{2}
47 1+3.65T+47T2 1 + 3.65T + 47T^{2}
53 1+0.215T+53T2 1 + 0.215T + 53T^{2}
59 1+2.78T+59T2 1 + 2.78T + 59T^{2}
61 1−9.03T+61T2 1 - 9.03T + 61T^{2}
67 1+7.66T+67T2 1 + 7.66T + 67T^{2}
71 1−4.90T+71T2 1 - 4.90T + 71T^{2}
73 1−15.5T+73T2 1 - 15.5T + 73T^{2}
79 1−9.43T+79T2 1 - 9.43T + 79T^{2}
83 1+4.09T+83T2 1 + 4.09T + 83T^{2}
89 1+0.418T+89T2 1 + 0.418T + 89T^{2}
97 1+7.11T+97T2 1 + 7.11T + 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

−10.53037548627621170919192519519, −9.738586663512547851786590594763, −8.192831151997316336892533342234, −6.76550769822222441770282968481, −6.22389698425433008111806872943, −5.50561498754813863688883229204, −4.76321301533434046171986553243, −4.00510973766594395050002517318, −2.17913938645567897664155777485, 0, 2.17913938645567897664155777485, 4.00510973766594395050002517318, 4.76321301533434046171986553243, 5.50561498754813863688883229204, 6.22389698425433008111806872943, 6.76550769822222441770282968481, 8.192831151997316336892533342234, 9.738586663512547851786590594763, 10.53037548627621170919192519519

Graph of the ZZ-function along the critical line