Properties

Label 2-637-1.1-c1-0-23
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  + 2.72·2-s − 1.34·3-s + 5.42·4-s + 2.18·5-s − 3.66·6-s + 9.33·8-s − 1.18·9-s + 5.96·10-s − 1.04·11-s − 7.30·12-s + 13-s − 2.94·15-s + 14.5·16-s − 5.29·17-s − 3.23·18-s + 0.756·19-s + 11.8·20-s − 2.85·22-s + 0.653·23-s − 12.5·24-s − 0.216·25-s + 2.72·26-s + 5.63·27-s − 3.10·29-s − 8.02·30-s + 1.02·31-s + 21.1·32-s + ⋯
L(s)  = 1  + 1.92·2-s − 0.777·3-s + 2.71·4-s + 0.978·5-s − 1.49·6-s + 3.30·8-s − 0.395·9-s + 1.88·10-s − 0.316·11-s − 2.10·12-s + 0.277·13-s − 0.760·15-s + 3.64·16-s − 1.28·17-s − 0.762·18-s + 0.173·19-s + 2.65·20-s − 0.609·22-s + 0.136·23-s − 2.56·24-s − 0.0432·25-s + 0.534·26-s + 1.08·27-s − 0.576·29-s − 1.46·30-s + 0.184·31-s + 3.73·32-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

\(L(1)\) \(\approx\) \(4.109322482\)
\(L(\frac12)\) \(\approx\) \(4.109322482\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.72T + 2T^{2} \)
3 \( 1 + 1.34T + 3T^{2} \)
5 \( 1 - 2.18T + 5T^{2} \)
11 \( 1 + 1.04T + 11T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 0.756T + 19T^{2} \)
23 \( 1 - 0.653T + 23T^{2} \)
29 \( 1 + 3.10T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 7.32T + 41T^{2} \)
43 \( 1 - 0.887T + 43T^{2} \)
47 \( 1 - 2.33T + 47T^{2} \)
53 \( 1 - 4.88T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + 6.60T + 71T^{2} \)
73 \( 1 + 8.28T + 73T^{2} \)
79 \( 1 - 2.14T + 79T^{2} \)
83 \( 1 + 6.66T + 83T^{2} \)
89 \( 1 + 5.76T + 89T^{2} \)
97 \( 1 + 2.88T + 97T^{2} \)
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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

−10.93551861639532281696511919677, −10.26753545585311700536443795958, −8.854394168562985916335001078001, −7.36247631916431728265807451508, −6.44745540255808152087135835720, −5.83501730523927416146340520122, −5.21836943092787905449818572111, −4.27510560068539662712211202578, −2.96031699784824758921161016485, −1.91451706145432172953967256008, 1.91451706145432172953967256008, 2.96031699784824758921161016485, 4.27510560068539662712211202578, 5.21836943092787905449818572111, 5.83501730523927416146340520122, 6.44745540255808152087135835720, 7.36247631916431728265807451508, 8.854394168562985916335001078001, 10.26753545585311700536443795958, 10.93551861639532281696511919677

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