Properties

Label 2-637-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.470·2-s − 2.24·3-s − 1.77·4-s − 0.529·5-s − 1.05·6-s − 1.77·8-s + 2.05·9-s − 0.249·10-s − 2.24·11-s + 4.00·12-s − 13-s + 1.19·15-s + 2.71·16-s + 1.30·17-s + 0.968·18-s + 1.47·19-s + 0.941·20-s − 1.05·22-s + 5.83·23-s + 4.00·24-s − 4.71·25-s − 0.470·26-s + 2.11·27-s + 5.22·29-s + 0.560·30-s + 7.02·31-s + 4.83·32-s + ⋯
L(s)  = 1  + 0.332·2-s − 1.29·3-s − 0.889·4-s − 0.236·5-s − 0.432·6-s − 0.628·8-s + 0.686·9-s − 0.0787·10-s − 0.678·11-s + 1.15·12-s − 0.277·13-s + 0.307·15-s + 0.679·16-s + 0.317·17-s + 0.228·18-s + 0.337·19-s + 0.210·20-s − 0.225·22-s + 1.21·23-s + 0.816·24-s − 0.943·25-s − 0.0923·26-s + 0.407·27-s + 0.969·29-s + 0.102·30-s + 1.26·31-s + 0.855·32-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: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6687979106\)
\(L(\frac12)\) \(\approx\) \(0.6687979106\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.470T + 2T^{2} \)
3 \( 1 + 2.24T + 3T^{2} \)
5 \( 1 + 0.529T + 5T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 - 5.83T + 23T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 + 2.36T + 37T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + 8.58T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 6.91T + 89T^{2} \)
97 \( 1 - 3.47T + 97T^{2} \)
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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

−10.53479542699030671970451914933, −9.983697240012248097291284913560, −8.858273253139612343567181957335, −7.936017168618540638475479885292, −6.81802789260794889648396775126, −5.76250169193704556090730717133, −5.13183137493496915966746099193, −4.36088514355770213159537729886, −3.00857864923386076986054653525, −0.71272129291482955258320604293, 0.71272129291482955258320604293, 3.00857864923386076986054653525, 4.36088514355770213159537729886, 5.13183137493496915966746099193, 5.76250169193704556090730717133, 6.81802789260794889648396775126, 7.936017168618540638475479885292, 8.858273253139612343567181957335, 9.983697240012248097291284913560, 10.53479542699030671970451914933

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