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Properties

Label 2-637-91.9-c1-0-14
Degree $2$
Conductor $637$
Sign $-0.803 - 0.595i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 2.26i)2-s − 2.61·3-s + (−2.42 + 4.20i)4-s + (−1.30 + 2.26i)5-s + (3.42 + 5.93i)6-s + 7.47·8-s + 3.85·9-s + 6.85·10-s + 1.85·11-s + (6.35 − 11.0i)12-s + (−2.5 + 2.59i)13-s + (3.42 − 5.93i)15-s + (−4.92 − 8.53i)16-s + (0.736 − 1.27i)17-s + (−5.04 − 8.73i)18-s − 1.85·19-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)2-s − 1.51·3-s + (−1.21 + 2.10i)4-s + (−0.585 + 1.01i)5-s + (1.39 + 2.42i)6-s + 2.64·8-s + 1.28·9-s + 2.16·10-s + 0.559·11-s + (1.83 − 3.17i)12-s + (−0.693 + 0.720i)13-s + (0.884 − 1.53i)15-s + (−1.23 − 2.13i)16-s + (0.178 − 0.309i)17-s + (−1.18 − 2.05i)18-s − 0.425·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (1.30 + 2.26i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 + (1.30 - 2.26i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
17 \( 1 + (-0.736 + 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.35 - 4.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.381 - 0.661i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.28 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.11 + 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.88 + 3.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.11 + 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + (-7.09 - 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (2.45 + 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.42 + 16.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.43243036520273762396641914601, −9.588013591285614976467569094392, −8.663908478121610016983128225278, −7.27421483754488196004475840244, −6.83450220255063357740806204950, −5.24514147000313573403109994682, −4.09147235014513230646258640690, −3.11221270724253022948343790276, −1.61714557940711332385785988550, 0, 1.00929791706032927283141065071, 4.47283315213456736615072293224, 4.97042761414765212222222217190, 5.97710591948254184475290006370, 6.50999378888106660827924594467, 7.62520217637335834017986954306, 8.242991401730279972782828780390, 9.217576369505501305152041262604, 10.04060797145740518792909287205

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