Properties

Label 2-637-1.1-c1-0-10
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  − 2.00·2-s + 1.75·3-s + 2.03·4-s + 0.905·5-s − 3.53·6-s − 0.0686·8-s + 0.0942·9-s − 1.81·10-s + 0.716·11-s + 3.57·12-s + 13-s + 1.59·15-s − 3.93·16-s + 2.35·17-s − 0.189·18-s + 6.63·19-s + 1.84·20-s − 1.43·22-s + 3.75·23-s − 0.120·24-s − 4.17·25-s − 2.00·26-s − 5.11·27-s + 3.25·29-s − 3.20·30-s + 1.57·31-s + 8.03·32-s + ⋯
L(s)  = 1  − 1.42·2-s + 1.01·3-s + 1.01·4-s + 0.405·5-s − 1.44·6-s − 0.0242·8-s + 0.0314·9-s − 0.575·10-s + 0.215·11-s + 1.03·12-s + 0.277·13-s + 0.411·15-s − 0.982·16-s + 0.570·17-s − 0.0446·18-s + 1.52·19-s + 0.411·20-s − 0.306·22-s + 0.783·23-s − 0.0246·24-s − 0.835·25-s − 0.393·26-s − 0.983·27-s + 0.604·29-s − 0.584·30-s + 0.282·31-s + 1.41·32-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

\(L(1)\) \(\approx\) \(1.130500812\)
\(L(\frac12)\) \(\approx\) \(1.130500812\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.00T + 2T^{2} \)
3 \( 1 - 1.75T + 3T^{2} \)
5 \( 1 - 0.905T + 5T^{2} \)
11 \( 1 - 0.716T + 11T^{2} \)
17 \( 1 - 2.35T + 17T^{2} \)
19 \( 1 - 6.63T + 19T^{2} \)
23 \( 1 - 3.75T + 23T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 - 1.57T + 31T^{2} \)
37 \( 1 - 5.20T + 37T^{2} \)
41 \( 1 - 4.92T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + 8.31T + 47T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 - 0.716T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 9.39T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 3.47T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 - 3.54T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 7.43T + 97T^{2} \)
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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

−10.10324994463521285756696217868, −9.549942403437567981855572351077, −8.953830049485826309312856779460, −8.080940900023863337763923179481, −7.56470950196379242090552853405, −6.44117932276499983603005377882, −5.15561657819510436261130487641, −3.57686451437127174993477717016, −2.46106698038607966999029540644, −1.17932930021618709396735525087, 1.17932930021618709396735525087, 2.46106698038607966999029540644, 3.57686451437127174993477717016, 5.15561657819510436261130487641, 6.44117932276499983603005377882, 7.56470950196379242090552853405, 8.080940900023863337763923179481, 8.953830049485826309312856779460, 9.549942403437567981855572351077, 10.10324994463521285756696217868

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