Properties

Label 2-637-1.1-c1-0-26
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  − 2.61·2-s + 2.23·3-s + 4.85·4-s − 2.23·5-s − 5.85·6-s − 7.47·8-s + 2.00·9-s + 5.85·10-s − 3·11-s + 10.8·12-s + 13-s − 5.00·15-s + 9.85·16-s − 1.47·17-s − 5.23·18-s − 3·19-s − 10.8·20-s + 7.85·22-s − 8.23·23-s − 16.7·24-s − 2.61·26-s − 2.23·27-s + 4.47·29-s + 13.0·30-s − 5·31-s − 10.8·32-s − 6.70·33-s + ⋯
L(s)  = 1  − 1.85·2-s + 1.29·3-s + 2.42·4-s − 0.999·5-s − 2.38·6-s − 2.64·8-s + 0.666·9-s + 1.85·10-s − 0.904·11-s + 3.13·12-s + 0.277·13-s − 1.29·15-s + 2.46·16-s − 0.357·17-s − 1.23·18-s − 0.688·19-s − 2.42·20-s + 1.67·22-s − 1.71·23-s − 3.41·24-s − 0.513·26-s − 0.430·27-s + 0.830·29-s + 2.38·30-s − 0.898·31-s − 1.91·32-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 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 7.47T + 47T^{2} \)
53 \( 1 + 7.47T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−9.926967043435727606751488411465, −9.073591398621732359811111073913, −8.279674204242944344175756530563, −7.979585550272571007696206837437, −7.27230104557089325010484293257, −6.07649062075562653452378841332, −4.09084762809047939404241567772, −2.90462871799594758035438569900, −1.95910020964794718294263088171, 0, 1.95910020964794718294263088171, 2.90462871799594758035438569900, 4.09084762809047939404241567772, 6.07649062075562653452378841332, 7.27230104557089325010484293257, 7.979585550272571007696206837437, 8.279674204242944344175756530563, 9.073591398621732359811111073913, 9.926967043435727606751488411465

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