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Properties

Label 2-637-13.4-c1-0-21
Degree $2$
Conductor $637$
Sign $-0.794 + 0.607i$
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  + (−1.20 − 0.692i)2-s + (−1.41 + 2.44i)3-s + (−0.0395 − 0.0685i)4-s − 0.518i·5-s + (3.39 − 1.95i)6-s + 2.88i·8-s + (−2.49 − 4.31i)9-s + (−0.359 + 0.622i)10-s + (1.40 + 0.812i)11-s + 0.223·12-s + (−1.42 + 3.31i)13-s + (1.26 + 0.733i)15-s + (1.91 − 3.32i)16-s + (−0.974 − 1.68i)17-s + 6.90i·18-s + (−2.15 + 1.24i)19-s + ⋯
L(s)  = 1  + (−0.848 − 0.490i)2-s + (−0.815 + 1.41i)3-s + (−0.0197 − 0.0342i)4-s − 0.232i·5-s + (1.38 − 0.799i)6-s + 1.01i·8-s + (−0.830 − 1.43i)9-s + (−0.113 + 0.196i)10-s + (0.424 + 0.244i)11-s + 0.0645·12-s + (−0.395 + 0.918i)13-s + (0.327 + 0.189i)15-s + (0.479 − 0.830i)16-s + (−0.236 − 0.409i)17-s + 1.62i·18-s + (−0.494 + 0.285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.794 + 0.607i$
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: \((2,\ 637,\ (\ :1/2),\ -0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00772375 - 0.0227989i\)
\(L(\frac12)\) \(\approx\) \(0.00772375 - 0.0227989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 3.31i)T \)
good2 \( 1 + (1.20 + 0.692i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.518iT - 5T^{2} \)
11 \( 1 + (-1.40 - 0.812i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.974 + 1.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.15 - 1.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.57 - 7.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.79iT - 31T^{2} \)
37 \( 1 + (8.85 + 5.11i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.64 + 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.51iT - 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (-5.37 + 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.73 + 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.25 + 4.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.50 - 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 0.982T + 79T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + (-10.4 - 6.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.82 - 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.20190334696250424671125291527, −9.383536739730392928850706201021, −9.163831299127219683415092324065, −7.903687922570253457729497292766, −6.48649889626505182148509261714, −5.42522490425985678718252604264, −4.69600577704211388938683414450, −3.73967751392240169360337710636, −1.91679580768653269681176508406, −0.02075371238265309940996740853, 1.32105192132306422578713666422, 2.98616468891736078289439379069, 4.68326208909710701250247832133, 6.03870700970991045221503031648, 6.71265536786903299536550854901, 7.24623767568179510942860862963, 8.332684333341369261831295585019, 8.683486878344473485494772161424, 10.23224222937341826202819475824, 10.71849413053351531522281592892

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