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Properties

Label 2-637-91.25-c1-0-31
Degree $2$
Conductor $637$
Sign $0.534 + 0.845i$
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  + (0.596 − 0.344i)2-s + (1.10 − 1.91i)3-s + (−0.762 + 1.32i)4-s + (2.78 − 1.60i)5-s − 1.52i·6-s + 2.42i·8-s + (−0.951 − 1.64i)9-s + (1.10 − 1.91i)10-s + (2.32 + 1.34i)11-s + (1.68 + 2.92i)12-s + (3.59 + 0.311i)13-s − 7.11i·15-s + (−0.688 − 1.19i)16-s + (1.79 − 3.11i)17-s + (−1.13 − 0.655i)18-s + (−7.40 + 4.27i)19-s + ⋯
L(s)  = 1  + (0.421 − 0.243i)2-s + (0.639 − 1.10i)3-s + (−0.381 + 0.660i)4-s + (1.24 − 0.718i)5-s − 0.622i·6-s + 0.858i·8-s + (−0.317 − 0.549i)9-s + (0.350 − 0.606i)10-s + (0.702 + 0.405i)11-s + (0.487 + 0.844i)12-s + (0.996 + 0.0862i)13-s − 1.83i·15-s + (−0.172 − 0.298i)16-s + (0.435 − 0.754i)17-s + (−0.267 − 0.154i)18-s + (−1.69 + 0.980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.534 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27828 - 1.25476i\)
\(L(\frac12)\) \(\approx\) \(2.27828 - 1.25476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 - 0.311i)T \)
good2 \( 1 + (-0.596 + 0.344i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.10 + 1.91i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.78 + 1.60i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.32 - 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.79 + 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.40 - 4.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.64 + 2.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + (5.05 + 2.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.40 - 1.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.755iT - 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 + (1.63 - 0.941i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.26 - 2.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.34 - 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.52 + 7.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.371 + 0.214i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.98iT - 71T^{2} \)
73 \( 1 + (-5.01 - 2.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.23 - 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + (4.64 - 2.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.62iT - 97T^{2} \)
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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

−10.36461069624077498896333327991, −9.306736041825090082685828630759, −8.588073681138836564934995966531, −8.057614352092371007125588945908, −6.82290809946435903054947474052, −5.99276317799880123007804202579, −4.82022825939508338434669775294, −3.70942389105735489062104277575, −2.32144941845840860516883321057, −1.53553638597325528279544524679, 1.76341207155737047724869555234, 3.33957640981997770097136362812, 4.09323290421757686952500220286, 5.24022019011269134943257378646, 6.19252779440871153895465114729, 6.67745280333874722121423314947, 8.588530934129813689498435023579, 9.056774501100472275088658210977, 9.938998539544106912702310623491, 10.46499306156585489213590767996

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