Properties

Label 2-637-1.1-c1-0-4
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  − 0.656·2-s − 0.204·3-s − 1.56·4-s − 1.35·5-s + 0.134·6-s + 2.34·8-s − 2.95·9-s + 0.892·10-s − 1.90·11-s + 0.321·12-s + 13-s + 0.278·15-s + 1.60·16-s + 3.56·17-s + 1.94·18-s + 0.985·19-s + 2.13·20-s + 1.25·22-s + 1.69·23-s − 0.479·24-s − 3.15·25-s − 0.656·26-s + 1.21·27-s + 6.54·29-s − 0.182·30-s + 7.69·31-s − 5.73·32-s + ⋯
L(s)  = 1  − 0.463·2-s − 0.118·3-s − 0.784·4-s − 0.608·5-s + 0.0548·6-s + 0.828·8-s − 0.986·9-s + 0.282·10-s − 0.574·11-s + 0.0927·12-s + 0.277·13-s + 0.0718·15-s + 0.400·16-s + 0.864·17-s + 0.457·18-s + 0.226·19-s + 0.477·20-s + 0.266·22-s + 0.353·23-s − 0.0978·24-s − 0.630·25-s − 0.128·26-s + 0.234·27-s + 1.21·29-s − 0.0333·30-s + 1.38·31-s − 1.01·32-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

\(L(1)\) \(\approx\) \(0.6866795523\)
\(L(\frac12)\) \(\approx\) \(0.6866795523\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.656T + 2T^{2} \)
3 \( 1 + 0.204T + 3T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
19 \( 1 - 0.985T + 19T^{2} \)
23 \( 1 - 1.69T + 23T^{2} \)
29 \( 1 - 6.54T + 29T^{2} \)
31 \( 1 - 7.69T + 31T^{2} \)
37 \( 1 + 2.02T + 37T^{2} \)
41 \( 1 - 9.88T + 41T^{2} \)
43 \( 1 + 3.16T + 43T^{2} \)
47 \( 1 - 7.76T + 47T^{2} \)
53 \( 1 - 0.354T + 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 9.05T + 71T^{2} \)
73 \( 1 + 7.13T + 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 + 6.89T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−10.43370309948356759355377065584, −9.745838140221454045526974486032, −8.633099350450562745822207012641, −8.193375193607149075719694659386, −7.34250280098881812049235129319, −5.94653348036128069079180912338, −5.07536770607284294125650802195, −4.01975644818778433237683685396, −2.84860105763173388855996623049, −0.77488385266128951688398560546, 0.77488385266128951688398560546, 2.84860105763173388855996623049, 4.01975644818778433237683685396, 5.07536770607284294125650802195, 5.94653348036128069079180912338, 7.34250280098881812049235129319, 8.193375193607149075719694659386, 8.633099350450562745822207012641, 9.745838140221454045526974486032, 10.43370309948356759355377065584

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