Properties

Label 2-637-1.1-c1-0-24
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  − 2·2-s + 2·4-s + 3·5-s − 3·9-s − 6·10-s − 6·11-s + 13-s − 4·16-s − 4·17-s + 6·18-s − 5·19-s + 6·20-s + 12·22-s + 3·23-s + 4·25-s − 2·26-s − 5·29-s + 3·31-s + 8·32-s + 8·34-s − 6·36-s − 4·37-s + 10·38-s + 6·41-s − 43-s − 12·44-s − 9·45-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.34·5-s − 9-s − 1.89·10-s − 1.80·11-s + 0.277·13-s − 16-s − 0.970·17-s + 1.41·18-s − 1.14·19-s + 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s − 0.392·26-s − 0.928·29-s + 0.538·31-s + 1.41·32-s + 1.37·34-s − 36-s − 0.657·37-s + 1.62·38-s + 0.937·41-s − 0.152·43-s − 1.80·44-s − 1.34·45-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 \( 1 + p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−10.06879161414549558260575138607, −9.232409444628192264737671791785, −8.568760068955747319631158439108, −7.85318737682082614131812951371, −6.67298853468060181727432085852, −5.78821104763479917640596143353, −4.82450768688560173606084932648, −2.72528535271701611265854270606, −1.93185778189567456463072050241, 0, 1.93185778189567456463072050241, 2.72528535271701611265854270606, 4.82450768688560173606084932648, 5.78821104763479917640596143353, 6.67298853468060181727432085852, 7.85318737682082614131812951371, 8.568760068955747319631158439108, 9.232409444628192264737671791785, 10.06879161414549558260575138607

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