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Properties

Label 2-637-13.3-c1-0-12
Degree $2$
Conductor $637$
Sign $0.872 - 0.488i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (1.5 + 2.59i)3-s + (0.500 − 0.866i)4-s + 3·5-s + (1.5 − 2.59i)6-s − 3·8-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)10-s + (1.5 + 2.59i)11-s + 3·12-s + (−1 + 3.46i)13-s + (4.5 + 7.79i)15-s + (0.500 + 0.866i)16-s + (1 − 1.73i)17-s + 6·18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.866 + 1.49i)3-s + (0.250 − 0.433i)4-s + 1.34·5-s + (0.612 − 1.06i)6-s − 1.06·8-s + (−1 + 1.73i)9-s + (−0.474 − 0.821i)10-s + (0.452 + 0.783i)11-s + 0.866·12-s + (−0.277 + 0.960i)13-s + (1.16 + 2.01i)15-s + (0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + 1.41·18-s + (0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02908 + 0.529771i\)
\(L(\frac12)\) \(\approx\) \(2.02908 + 0.529771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.16951030776102695258004177625, −9.842038728476668612465377262783, −9.324793331059242290795139831263, −8.699385849675814534459482604763, −7.09393507978738512689455294188, −5.92365999219625160161204773627, −5.01777702992843472652753311492, −3.95362846861699277713048297047, −2.60732321400380514698232162966, −1.91499391943724190431809738986, 1.31299646496539755696952707505, 2.54814639524729726420235275367, 3.32499173102226489993322310894, 5.68406590572197503629335054721, 6.21532568445604949993832185577, 7.07219297765751453787319442940, 7.84105277319775683298564600351, 8.598070929724526489489679568355, 9.188914032787288603776484306400, 10.27962439364336452354976674814

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