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Properties

Label 2-637-91.81-c1-0-13
Degree $2$
Conductor $637$
Sign $0.939 - 0.342i$
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  + (−1.21 + 2.10i)2-s − 0.753·3-s + (−1.95 − 3.39i)4-s + (0.170 + 0.295i)5-s + (0.916 − 1.58i)6-s + 4.65·8-s − 2.43·9-s − 0.830·10-s − 2.43·11-s + (1.47 + 2.55i)12-s + (−2.50 + 2.59i)13-s + (−0.128 − 0.222i)15-s + (−1.74 + 3.02i)16-s + (−0.974 − 1.68i)17-s + (2.95 − 5.12i)18-s + 6.29·19-s + ⋯
L(s)  = 1  + (−0.859 + 1.48i)2-s − 0.435·3-s + (−0.978 − 1.69i)4-s + (0.0763 + 0.132i)5-s + (0.374 − 0.647i)6-s + 1.64·8-s − 0.810·9-s − 0.262·10-s − 0.733·11-s + (0.425 + 0.737i)12-s + (−0.693 + 0.720i)13-s + (−0.0332 − 0.0575i)15-s + (−0.437 + 0.757i)16-s + (−0.236 − 0.409i)17-s + (0.697 − 1.20i)18-s + 1.44·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461474 + 0.0814343i\)
\(L(\frac12)\) \(\approx\) \(0.461474 + 0.0814343i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 - 2.59i)T \)
good2 \( 1 + (1.21 - 2.10i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.753T + 3T^{2} \)
5 \( 1 + (-0.170 - 0.295i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.43T + 11T^{2} \)
17 \( 1 + (0.974 + 1.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.29T + 19T^{2} \)
23 \( 1 + (-1.84 + 3.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.22 + 3.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.987 - 1.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.26 - 10.8i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.20 + 7.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.50 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.746 - 1.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.313 - 0.542i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 + 5.59T + 67T^{2} \)
71 \( 1 + (4.74 - 8.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.95 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.23 + 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 + (-6.22 + 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.13 + 8.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.33748236179586269192298144554, −9.483113236605014283306307954492, −8.815036782479067317439091884169, −7.82603199682362159638732057044, −7.17824343932956534213094136016, −6.26236972055888016217106639848, −5.47483197936505955306016065268, −4.68577219376450336121145293966, −2.67951226208558297893152438853, −0.43554863557981812431947576478, 1.07127041897972966231277231557, 2.62962589377810023951117722675, 3.34831203611429770078683368418, 4.92871286233877608047780341961, 5.77225416052931937572521627278, 7.42211906390402054450233642412, 8.129950037824926757744425502170, 9.175828095089196383112123791038, 9.723173792031318309872892210295, 10.73550586385557232755902777308

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