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Properties

Label 2-637-13.3-c1-0-37
Degree $2$
Conductor $637$
Sign $0.367 - 0.929i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 1.99i)2-s + (−1.08 − 1.87i)3-s + (−1.65 + 2.86i)4-s + 2.16·5-s + (−2.49 + 4.32i)6-s + 2.99·8-s + (−0.848 + 1.46i)9-s + (−2.49 − 4.32i)10-s + (−2.45 − 4.25i)11-s + 7.15·12-s + (−1.41 − 3.31i)13-s + (−2.34 − 4.06i)15-s + (−0.151 − 0.262i)16-s + (−3.57 + 6.19i)17-s + 3.90·18-s + (−1.08 + 1.87i)19-s + ⋯
L(s)  = 1  + (−0.814 − 1.41i)2-s + (−0.625 − 1.08i)3-s + (−0.825 + 1.43i)4-s + 0.969·5-s + (−1.01 + 1.76i)6-s + 1.06·8-s + (−0.282 + 0.489i)9-s + (−0.789 − 1.36i)10-s + (−0.739 − 1.28i)11-s + 2.06·12-s + (−0.391 − 0.920i)13-s + (−0.606 − 1.05i)15-s + (−0.0378 − 0.0655i)16-s + (−0.868 + 1.50i)17-s + 0.921·18-s + (−0.248 + 0.430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.367 - 0.929i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269291 + 0.183073i\)
\(L(\frac12)\) \(\approx\) \(0.269291 + 0.183073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 3.31i)T \)
good2 \( 1 + (1.15 + 1.99i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.08 + 1.87i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.16T + 5T^{2} \)
11 \( 1 + (2.45 + 4.25i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.57 - 6.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.08 - 1.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.302 - 0.524i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.15 - 1.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.15T + 31T^{2} \)
37 \( 1 + (4.30 + 7.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.99 - 8.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.25 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.51T + 47T^{2} \)
53 \( 1 - 2.39T + 53T^{2} \)
59 \( 1 + (-1.41 + 2.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.16 - 3.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.33T + 73T^{2} \)
79 \( 1 + 6.60T + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + (3.25 + 5.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.83 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.26222897437973536190597119266, −9.097450948511925333605159400527, −8.373809881170775037067204240049, −7.44394832846832267683733722316, −6.04833805952108573399105828853, −5.65502845650550796114012482173, −3.68960547837397041636009258577, −2.42441246894867687522291625691, −1.56284211725174995451881222743, −0.24180487227083459192103064091, 2.27767983567120718258852681783, 4.56728148578927855568062542163, 5.02103384332134758367868695578, 5.94202421540598208808864746187, 6.92498480371723243229774325658, 7.53544377308828971296252325705, 8.994608713065853507283245242052, 9.487778786814506818322621036381, 9.978520127920313271985474553002, 10.83614840734325901527247090856

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