Properties

Label 2-637-1.1-c1-0-37
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

−10.07084765051358713543920425978, −9.259730413329108464669848282618, −8.427602051723064213968857703075, −7.58221538537525761723731254609, −6.13619288958387853257590270753, −5.56220221892590651265131476874, −4.57403153778540150751986491981, −3.52729931218747647603184716659, −2.43258577292117985821674468072, 0, 2.43258577292117985821674468072, 3.52729931218747647603184716659, 4.57403153778540150751986491981, 5.56220221892590651265131476874, 6.13619288958387853257590270753, 7.58221538537525761723731254609, 8.427602051723064213968857703075, 9.259730413329108464669848282618, 10.07084765051358713543920425978

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