Properties

Label 2-637-1.1-c1-0-36
Degree $2$
Conductor $637$
Sign $-1$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 3·9-s − 3·11-s − 13-s − 16-s + 7·17-s − 3·18-s − 7·19-s − 3·22-s − 6·23-s − 5·25-s − 26-s − 5·29-s + 5·32-s + 7·34-s + 3·36-s + 8·37-s − 7·38-s + 2·43-s + 3·44-s − 6·46-s + 7·47-s − 5·50-s + 52-s − 3·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s − 0.904·11-s − 0.277·13-s − 1/4·16-s + 1.69·17-s − 0.707·18-s − 1.60·19-s − 0.639·22-s − 1.25·23-s − 25-s − 0.196·26-s − 0.928·29-s + 0.883·32-s + 1.20·34-s + 1/2·36-s + 1.31·37-s − 1.13·38-s + 0.304·43-s + 0.452·44-s − 0.884·46-s + 1.02·47-s − 0.707·50-s + 0.138·52-s − 0.412·53-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: yes
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 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−10.11909095240905822343036232182, −9.332541423959734621527612295695, −8.264528127864852069730211246497, −7.71886042427351779334180674638, −5.97764942510297289458867575451, −5.71244157014039939594521038014, −4.53493125422219918903828313314, −3.54025539217060415386292648543, −2.43005796671753573111852467369, 0, 2.43005796671753573111852467369, 3.54025539217060415386292648543, 4.53493125422219918903828313314, 5.71244157014039939594521038014, 5.97764942510297289458867575451, 7.71886042427351779334180674638, 8.264528127864852069730211246497, 9.332541423959734621527612295695, 10.11909095240905822343036232182

Graph of the $Z$-function along the critical line