Properties

Label 2-637-1.1-c1-0-20
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  − 1.17·2-s + 3.35·3-s − 0.616·4-s + 3.14·5-s − 3.94·6-s + 3.07·8-s + 8.22·9-s − 3.70·10-s − 0.773·11-s − 2.06·12-s + 13-s + 10.5·15-s − 2.38·16-s − 5.75·17-s − 9.67·18-s − 1.22·19-s − 1.94·20-s + 0.909·22-s − 2.99·23-s + 10.3·24-s + 4.91·25-s − 1.17·26-s + 17.5·27-s − 2.46·29-s − 12.4·30-s − 6.13·31-s − 3.34·32-s + ⋯
L(s)  = 1  − 0.831·2-s + 1.93·3-s − 0.308·4-s + 1.40·5-s − 1.60·6-s + 1.08·8-s + 2.74·9-s − 1.17·10-s − 0.233·11-s − 0.596·12-s + 0.277·13-s + 2.72·15-s − 0.596·16-s − 1.39·17-s − 2.28·18-s − 0.280·19-s − 0.434·20-s + 0.193·22-s − 0.623·23-s + 2.10·24-s + 0.982·25-s − 0.230·26-s + 3.37·27-s − 0.458·29-s − 2.26·30-s − 1.10·31-s − 0.591·32-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

\(L(1)\) \(\approx\) \(2.081214073\)
\(L(\frac12)\) \(\approx\) \(2.081214073\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.17T + 2T^{2} \)
3 \( 1 - 3.35T + 3T^{2} \)
5 \( 1 - 3.14T + 5T^{2} \)
11 \( 1 + 0.773T + 11T^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 + 2.99T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 6.13T + 31T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 + 2.55T + 41T^{2} \)
43 \( 1 + 2.73T + 43T^{2} \)
47 \( 1 - 5.37T + 47T^{2} \)
53 \( 1 + 9.79T + 53T^{2} \)
59 \( 1 - 2.50T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 4.32T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 5.17T + 73T^{2} \)
79 \( 1 + 0.542T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 - 9.23T + 89T^{2} \)
97 \( 1 + 1.26T + 97T^{2} \)
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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

−10.13922704394775975336900034236, −9.331543853372087150861870829455, −9.091908285696682337428165960129, −8.252003237231177130657331323688, −7.46360338335008311325997787584, −6.38482276822183894928856597019, −4.84002741993096800399995300541, −3.78307621201181422753251217693, −2.35237178692968263429215304691, −1.68756983034729400615022091457, 1.68756983034729400615022091457, 2.35237178692968263429215304691, 3.78307621201181422753251217693, 4.84002741993096800399995300541, 6.38482276822183894928856597019, 7.46360338335008311325997787584, 8.252003237231177130657331323688, 9.091908285696682337428165960129, 9.331543853372087150861870829455, 10.13922704394775975336900034236

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