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Properties

Label 2-637-7.4-c1-0-9
Degree $2$
Conductor $637$
Sign $0.900 + 0.435i$
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  + (−0.916 − 1.58i)2-s + (0.426 − 0.739i)3-s + (−0.678 + 1.17i)4-s + (1.31 + 2.27i)5-s − 1.56·6-s − 1.17·8-s + (1.13 + 1.96i)9-s + (2.40 − 4.16i)10-s + (−1.63 + 2.82i)11-s + (0.579 + 1.00i)12-s − 13-s + 2.24·15-s + (2.43 + 4.21i)16-s + (2.26 − 3.92i)17-s + (2.08 − 3.60i)18-s + (2.03 + 3.52i)19-s + ⋯
L(s)  = 1  + (−0.647 − 1.12i)2-s + (0.246 − 0.426i)3-s + (−0.339 + 0.587i)4-s + (0.587 + 1.01i)5-s − 0.638·6-s − 0.416·8-s + (0.378 + 0.655i)9-s + (0.760 − 1.31i)10-s + (−0.492 + 0.852i)11-s + (0.167 + 0.289i)12-s − 0.277·13-s + 0.578·15-s + (0.609 + 1.05i)16-s + (0.549 − 0.951i)17-s + (0.490 − 0.849i)18-s + (0.466 + 0.807i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17203 - 0.268849i\)
\(L(\frac12)\) \(\approx\) \(1.17203 - 0.268849i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (0.916 + 1.58i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.426 + 0.739i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.31 - 2.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.63 - 2.82i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.26 + 3.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.03 - 3.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.26 - 3.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 + (1.40 - 2.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.02 - 8.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.84T + 41T^{2} \)
43 \( 1 - 9.72T + 43T^{2} \)
47 \( 1 + (4.72 + 8.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.63 - 4.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.28 - 2.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.59 + 9.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.990 + 1.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + (-6.06 + 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.95 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (-5.33 - 9.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−10.32819669575053664021743006741, −9.957086895914491394901900584745, −9.173483399313699591245835113408, −7.80483606241743217806724968718, −7.27947256766699109440101455070, −6.12274012638774327137471561566, −4.92883400183845616359848286787, −3.21692710915125314386719427011, −2.46126647926004148100034601654, −1.51319465825121362186922412846, 0.852161085668645886143658224213, 2.91324824150051318588768221609, 4.31493449630865995151001180604, 5.50737238454143948951642487695, 6.08881936441545104525950119507, 7.24668520789886153936578504501, 8.120811592994831253168338391463, 9.013066516742311813397223385489, 9.283492069118808248861792691999, 10.24020704069409772822035908133

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