Properties

Label 2-637-1.1-c1-0-31
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  + 2.34·2-s + 1.14·3-s + 3.48·4-s + 1.34·5-s + 2.68·6-s + 3.48·8-s − 1.68·9-s + 3.14·10-s + 1.14·11-s + 4.00·12-s − 13-s + 1.53·15-s + 1.19·16-s − 5.83·17-s − 3.94·18-s + 3.34·19-s + 4.68·20-s + 2.68·22-s − 3.17·23-s + 4.00·24-s − 3.19·25-s − 2.34·26-s − 5.37·27-s + 10.4·29-s + 3.60·30-s − 1.63·31-s − 4.17·32-s + ⋯
L(s)  = 1  + 1.65·2-s + 0.661·3-s + 1.74·4-s + 0.600·5-s + 1.09·6-s + 1.23·8-s − 0.561·9-s + 0.994·10-s + 0.345·11-s + 1.15·12-s − 0.277·13-s + 0.397·15-s + 0.299·16-s − 1.41·17-s − 0.930·18-s + 0.766·19-s + 1.04·20-s + 0.572·22-s − 0.662·23-s + 0.816·24-s − 0.639·25-s − 0.459·26-s − 1.03·27-s + 1.94·29-s + 0.658·30-s − 0.293·31-s − 0.738·32-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

\(L(1)\) \(\approx\) \(4.517336931\)
\(L(\frac12)\) \(\approx\) \(4.517336931\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.34T + 2T^{2} \)
3 \( 1 - 1.14T + 3T^{2} \)
5 \( 1 - 1.34T + 5T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
17 \( 1 + 5.83T + 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 + 3.17T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 - 8.51T + 37T^{2} \)
41 \( 1 - 0.292T + 41T^{2} \)
43 \( 1 + 8.15T + 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + 0.782T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 6.10T + 67T^{2} \)
71 \( 1 - 1.53T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 0.882T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−10.88133094215117859094217182785, −9.697945126639058778520668004483, −8.873697915208596104311527575212, −7.78625288877059975270319363518, −6.59010937519673736239124424302, −5.97417129810271293235761021520, −4.94775821718714729055606283018, −4.02691821596254686839092015711, −2.91868243165336358882903193607, −2.13648582623965500768110420761, 2.13648582623965500768110420761, 2.91868243165336358882903193607, 4.02691821596254686839092015711, 4.94775821718714729055606283018, 5.97417129810271293235761021520, 6.59010937519673736239124424302, 7.78625288877059975270319363518, 8.873697915208596104311527575212, 9.697945126639058778520668004483, 10.88133094215117859094217182785

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