Properties

Label 2-637-1.1-c1-0-33
Degree $2$
Conductor $637$
Sign $-1$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

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

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

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 - T \)
good2 \( 1 - 1.86T + 2T^{2} \)
3 \( 1 + 3.34T + 3T^{2} \)
5 \( 1 - 0.866T + 5T^{2} \)
11 \( 1 + 3.86T + 11T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + 5.24T + 23T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 - 7.56T + 31T^{2} \)
37 \( 1 + 4.83T + 37T^{2} \)
41 \( 1 + 4.06T + 41T^{2} \)
43 \( 1 - 4.03T + 43T^{2} \)
47 \( 1 + 3.65T + 47T^{2} \)
53 \( 1 + 0.215T + 53T^{2} \)
59 \( 1 + 2.78T + 59T^{2} \)
61 \( 1 - 9.03T + 61T^{2} \)
67 \( 1 + 7.66T + 67T^{2} \)
71 \( 1 - 4.90T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 - 9.43T + 79T^{2} \)
83 \( 1 + 4.09T + 83T^{2} \)
89 \( 1 + 0.418T + 89T^{2} \)
97 \( 1 + 7.11T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line