Properties

Label 2-637-1.1-c1-0-18
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s − 2.23·3-s − 1.85·4-s + 2.23·5-s + 0.854·6-s + 1.47·8-s + 2.00·9-s − 0.854·10-s − 3·11-s + 4.14·12-s + 13-s − 5.00·15-s + 3.14·16-s + 7.47·17-s − 0.763·18-s − 3·19-s − 4.14·20-s + 1.14·22-s − 3.76·23-s − 3.29·24-s − 0.381·26-s + 2.23·27-s − 4.47·29-s + 1.90·30-s − 5·31-s − 4.14·32-s + 6.70·33-s + ⋯
L(s)  = 1  − 0.270·2-s − 1.29·3-s − 0.927·4-s + 0.999·5-s + 0.348·6-s + 0.520·8-s + 0.666·9-s − 0.270·10-s − 0.904·11-s + 1.19·12-s + 0.277·13-s − 1.29·15-s + 0.786·16-s + 1.81·17-s − 0.180·18-s − 0.688·19-s − 0.927·20-s + 0.244·22-s − 0.784·23-s − 0.671·24-s − 0.0749·26-s + 0.430·27-s − 0.830·29-s + 0.348·30-s − 0.898·31-s − 0.732·32-s + 1.16·33-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: \(1\)
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 \( 1 + 0.381T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 8.70T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 - 1.47T + 53T^{2} \)
59 \( 1 + 7.47T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2.23T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−10.26813337319332781669343398813, −9.535584811349761561598349820299, −8.460241464774015638621155340295, −7.53959975314164666285963391409, −6.20773811119268649499049361066, −5.48987176006989565595941322782, −5.02887787297504592985778238416, −3.55744827318528851282072200960, −1.63436554889716791348719925216, 0, 1.63436554889716791348719925216, 3.55744827318528851282072200960, 5.02887787297504592985778238416, 5.48987176006989565595941322782, 6.20773811119268649499049361066, 7.53959975314164666285963391409, 8.460241464774015638621155340295, 9.535584811349761561598349820299, 10.26813337319332781669343398813

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