Properties

Label 2-637-1.1-c1-0-3
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  − 0.195·2-s + 0.259·3-s − 1.96·4-s − 3.93·5-s − 0.0508·6-s + 0.775·8-s − 2.93·9-s + 0.769·10-s + 4.50·11-s − 0.509·12-s − 13-s − 1.02·15-s + 3.77·16-s + 2.28·17-s + 0.573·18-s + 1.78·19-s + 7.71·20-s − 0.881·22-s + 1.74·23-s + 0.201·24-s + 10.4·25-s + 0.195·26-s − 1.54·27-s + 1.65·29-s + 0.199·30-s − 5.60·31-s − 2.28·32-s + ⋯
L(s)  = 1  − 0.138·2-s + 0.149·3-s − 0.980·4-s − 1.75·5-s − 0.0207·6-s + 0.274·8-s − 0.977·9-s + 0.243·10-s + 1.35·11-s − 0.147·12-s − 0.277·13-s − 0.263·15-s + 0.942·16-s + 0.553·17-s + 0.135·18-s + 0.410·19-s + 1.72·20-s − 0.187·22-s + 0.362·23-s + 0.0411·24-s + 2.09·25-s + 0.0383·26-s − 0.296·27-s + 0.306·29-s + 0.0364·30-s − 1.00·31-s − 0.404·32-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

\(L(1)\) \(\approx\) \(0.7345528446\)
\(L(\frac12)\) \(\approx\) \(0.7345528446\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.195T + 2T^{2} \)
3 \( 1 - 0.259T + 3T^{2} \)
5 \( 1 + 3.93T + 5T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
17 \( 1 - 2.28T + 17T^{2} \)
19 \( 1 - 1.78T + 19T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 - 1.65T + 29T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 - 7.14T + 37T^{2} \)
41 \( 1 - 8.11T + 41T^{2} \)
43 \( 1 - 6.81T + 43T^{2} \)
47 \( 1 + 3.54T + 47T^{2} \)
53 \( 1 - 3.28T + 53T^{2} \)
59 \( 1 + 4.50T + 59T^{2} \)
61 \( 1 + 7.54T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 9.54T + 71T^{2} \)
73 \( 1 + 1.08T + 73T^{2} \)
79 \( 1 - 0.791T + 79T^{2} \)
83 \( 1 - 7.14T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 8.81T + 97T^{2} \)
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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

−10.71508014006865188973121492234, −9.316498203483301759037707426538, −8.935551232213404303432312844269, −7.939173297416924679370484539626, −7.43815406640588316207137272427, −6.03733500914981787855237251718, −4.78668944615605967156322335428, −3.94866557361769110104162032915, −3.19708611540627795138923656606, −0.75923054352271009404722163639, 0.75923054352271009404722163639, 3.19708611540627795138923656606, 3.94866557361769110104162032915, 4.78668944615605967156322335428, 6.03733500914981787855237251718, 7.43815406640588316207137272427, 7.939173297416924679370484539626, 8.935551232213404303432312844269, 9.316498203483301759037707426538, 10.71508014006865188973121492234

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