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Properties

Label 2-637-13.4-c1-0-1
Degree $2$
Conductor $637$
Sign $0.376 - 0.926i$
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  + (−1.12 − 0.651i)2-s + (−0.134 + 0.233i)3-s + (−0.150 − 0.260i)4-s − 1.56i·5-s + (0.304 − 0.175i)6-s + 2.99i·8-s + (1.46 + 2.53i)9-s + (−1.02 + 1.77i)10-s + (−5.31 − 3.06i)11-s + 0.0810·12-s + (0.450 + 3.57i)13-s + (0.366 + 0.211i)15-s + (1.65 − 2.86i)16-s + (0.626 + 1.08i)17-s − 3.81i·18-s + (−6.32 + 3.65i)19-s + ⋯
L(s)  = 1  + (−0.798 − 0.460i)2-s + (−0.0779 + 0.134i)3-s + (−0.0751 − 0.130i)4-s − 0.701i·5-s + (0.124 − 0.0718i)6-s + 1.06i·8-s + (0.487 + 0.844i)9-s + (−0.323 + 0.559i)10-s + (−1.60 − 0.924i)11-s + 0.0234·12-s + (0.124 + 0.992i)13-s + (0.0946 + 0.0546i)15-s + (0.413 − 0.716i)16-s + (0.151 + 0.263i)17-s − 0.899i·18-s + (−1.45 + 0.837i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359931 + 0.242344i\)
\(L(\frac12)\) \(\approx\) \(0.359931 + 0.242344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.450 - 3.57i)T \)
good2 \( 1 + (1.12 + 0.651i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.134 - 0.233i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.56iT - 5T^{2} \)
11 \( 1 + (5.31 + 3.06i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.626 - 1.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.32 - 3.65i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 1.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0179 + 0.0311i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.09iT - 31T^{2} \)
37 \( 1 + (5.86 + 3.38i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.08 - 4.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.07 - 5.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 + 0.525T + 53T^{2} \)
59 \( 1 + (9.29 - 5.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.48 - 2.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.67 - 2.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.0892 + 0.0515i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.28iT - 73T^{2} \)
79 \( 1 + 4.57T + 79T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (-7.96 - 4.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.12 - 3.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.74477763399461535630650795336, −10.02967750972634129206207501498, −8.906227771628674970457829441423, −8.421063674336465512827743936728, −7.63022161118456206499886355867, −6.10966947911450122964278539755, −5.15467114566989862756170286293, −4.40073689021048364805624929121, −2.60145043833574633925911339554, −1.42089462990761544327920634086, 0.32077457341127796169180483958, 2.48856494372662421691522146806, 3.67810584659595819177391170833, 4.94120552748188595615592681997, 6.26431634264982696495829216518, 7.14679169841705336109004777607, 7.63986377685949049750230853092, 8.594922963498085374028902308270, 9.544737230817692623168403467798, 10.31480229550842968816310568627

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