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Properties

Label 2-630-315.178-c1-0-30
Degree $2$
Conductor $630$
Sign $0.670 + 0.741i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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  + (0.707 − 0.707i)2-s + (1.64 + 0.551i)3-s − 1.00i·4-s + (−0.512 − 2.17i)5-s + (1.55 − 0.771i)6-s + (2.17 + 1.50i)7-s + (−0.707 − 0.707i)8-s + (2.39 + 1.81i)9-s + (−1.90 − 1.17i)10-s + (1.02 + 1.76i)11-s + (0.551 − 1.64i)12-s + (−2.73 − 0.733i)13-s + (2.60 − 0.479i)14-s + (0.358 − 3.85i)15-s − 1.00·16-s + (7.04 − 1.88i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.948 + 0.318i)3-s − 0.500i·4-s + (−0.228 − 0.973i)5-s + (0.633 − 0.314i)6-s + (0.823 + 0.567i)7-s + (−0.250 − 0.250i)8-s + (0.797 + 0.603i)9-s + (−0.601 − 0.372i)10-s + (0.307 + 0.532i)11-s + (0.159 − 0.474i)12-s + (−0.758 − 0.203i)13-s + (0.695 − 0.128i)14-s + (0.0926 − 0.995i)15-s − 0.250·16-s + (1.70 − 0.457i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.670 + 0.741i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50260 - 1.11137i\)
\(L(\frac12)\) \(\approx\) \(2.50260 - 1.11137i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.64 - 0.551i)T \)
5 \( 1 + (0.512 + 2.17i)T \)
7 \( 1 + (-2.17 - 1.50i)T \)
good11 \( 1 + (-1.02 - 1.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.73 + 0.733i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-7.04 + 1.88i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.14 + 3.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.40 - 0.375i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.89 - 2.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.04iT - 31T^{2} \)
37 \( 1 + (10.6 + 2.85i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.89 - 4.55i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.17 - 1.65i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.768 - 0.768i)T + 47iT^{2} \)
53 \( 1 + (-2.48 + 0.665i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 - 8.54T + 59T^{2} \)
61 \( 1 - 12.7iT - 61T^{2} \)
67 \( 1 + (-5.30 + 5.30i)T - 67iT^{2} \)
71 \( 1 + 7.95T + 71T^{2} \)
73 \( 1 + (-3.52 - 13.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 - 2.57iT - 79T^{2} \)
83 \( 1 + (0.288 + 1.07i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.542 + 0.939i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.04 + 0.548i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.28007904722054167670938222732, −9.657972383018307100266184632190, −8.781041061547526974314668530354, −8.074261011451641099606720163229, −7.13169772025563364484737337970, −5.34937072125838568577894258437, −4.87377877503210260804881626137, −3.88254054615406172698447830744, −2.62917766579093399755263127640, −1.49126814168653915786645373935, 1.80693733986892788198521779081, 3.27769881150377369036878771060, 3.85742375524663979840207348263, 5.19271388169248333744632663324, 6.51001385757495090574262791485, 7.18046869915988873928596132496, 8.026086077605581290399671802269, 8.491746738369854470989317130606, 10.02227615364942637170355527696, 10.49063938388804297086168183894

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