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Properties

Label 2-637-13.3-c1-0-13
Degree $2$
Conductor $637$
Sign $0.182 + 0.983i$
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  + (−1.11 − 1.92i)2-s + (0.274 + 0.475i)3-s + (−1.46 + 2.53i)4-s + 4.22·5-s + (0.610 − 1.05i)6-s + 2.06·8-s + (1.34 − 2.33i)9-s + (−4.68 − 8.11i)10-s + (0.274 + 0.475i)11-s − 1.60·12-s + (2.95 + 2.06i)13-s + (1.15 + 2.00i)15-s + (0.640 + 1.10i)16-s + (−1.18 + 2.06i)17-s − 5.98·18-s + (−1.80 + 3.12i)19-s + ⋯
L(s)  = 1  + (−0.784 − 1.35i)2-s + (0.158 + 0.274i)3-s + (−0.732 + 1.26i)4-s + 1.88·5-s + (0.249 − 0.431i)6-s + 0.728·8-s + (0.449 − 0.778i)9-s + (−1.48 − 2.56i)10-s + (0.0828 + 0.143i)11-s − 0.464·12-s + (0.820 + 0.571i)13-s + (0.299 + 0.518i)15-s + (0.160 + 0.277i)16-s + (−0.288 + 0.499i)17-s − 1.41·18-s + (−0.414 + 0.717i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.182 + 0.983i$
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: \((2,\ 637,\ (\ :1/2),\ 0.182 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12695 - 0.936942i\)
\(L(\frac12)\) \(\approx\) \(1.12695 - 0.936942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.95 - 2.06i)T \)
good2 \( 1 + (1.11 + 1.92i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.274 - 0.475i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 4.22T + 5T^{2} \)
11 \( 1 + (-0.274 - 0.475i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.18 - 2.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.80 - 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.90 + 5.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.79 - 3.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.14T + 31T^{2} \)
37 \( 1 + (-0.164 - 0.285i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.14 + 5.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.61 - 2.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.20T + 47T^{2} \)
53 \( 1 - 2.65T + 53T^{2} \)
59 \( 1 + (0.903 - 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.304 + 0.527i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.18 + 8.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.90T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + (3.73 + 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.42 - 5.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.23470440990049145771107424319, −9.780096745736859697311494580054, −8.957269919507315684416197774175, −8.467568660967450255568681215701, −6.55552417241104523147958712391, −6.08240682698338020889921474312, −4.50014869647562525458607218141, −3.32774525822762247221761244566, −2.11903740020935137097550626141, −1.34441194758869814520385641349, 1.36042780716431300117501071694, 2.68407384375106487771147531526, 4.86824249175296855311612314387, 5.70994638006276909629941211023, 6.39001476357821859136531521931, 7.13722358029534888526263969338, 8.223833943259730379668339251092, 8.830272897243375022351860276420, 9.854488093200245195864927060336, 10.14293370435872133484891104272

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