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Properties

Label 2-630-315.194-c1-0-12
Degree $2$
Conductor $630$
Sign $0.951 - 0.306i$
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  + 2-s + (−1.70 + 0.304i)3-s + 4-s + (−1.95 − 1.08i)5-s + (−1.70 + 0.304i)6-s + (−2.42 + 1.06i)7-s + 8-s + (2.81 − 1.03i)9-s + (−1.95 − 1.08i)10-s + (3.64 + 2.10i)11-s + (−1.70 + 0.304i)12-s + (0.915 − 1.58i)13-s + (−2.42 + 1.06i)14-s + (3.66 + 1.24i)15-s + 16-s + (4.31 − 2.48i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.984 + 0.176i)3-s + 0.5·4-s + (−0.875 − 0.483i)5-s + (−0.696 + 0.124i)6-s + (−0.915 + 0.402i)7-s + 0.353·8-s + (0.938 − 0.346i)9-s + (−0.619 − 0.341i)10-s + (1.10 + 0.635i)11-s + (−0.492 + 0.0880i)12-s + (0.253 − 0.439i)13-s + (−0.647 + 0.284i)14-s + (0.946 + 0.321i)15-s + 0.250·16-s + (1.04 − 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.951 - 0.306i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43218 + 0.224744i\)
\(L(\frac12)\) \(\approx\) \(1.43218 + 0.224744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.70 - 0.304i)T \)
5 \( 1 + (1.95 + 1.08i)T \)
7 \( 1 + (2.42 - 1.06i)T \)
good11 \( 1 + (-3.64 - 2.10i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.915 + 1.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.31 + 2.48i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.39 - 3.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.26 - 7.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.68 - 1.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.46iT - 31T^{2} \)
37 \( 1 + (7.05 + 4.07i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.675 - 1.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.83 + 5.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.03iT - 47T^{2} \)
53 \( 1 + (-2.99 - 5.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.309T + 59T^{2} \)
61 \( 1 - 9.51iT - 61T^{2} \)
67 \( 1 + 10.1iT - 67T^{2} \)
71 \( 1 - 4.09iT - 71T^{2} \)
73 \( 1 + (-3.53 - 6.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.30T + 79T^{2} \)
83 \( 1 + (5.35 - 3.09i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0685 - 0.118i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.10 - 3.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.93005125240343708508476575420, −9.722676431486139079748802149217, −9.234538571009964687492518957454, −7.53528789983136022590155591443, −7.09067482354309060411397724613, −5.75938121332981179399357055767, −5.33016839899127876934331610149, −4.00103372467684352116121846750, −3.38872171756717890926474959086, −1.14605025857697197355360989526, 0.962264584789276322672146492681, 3.17723849723454466116286513327, 3.90729190251436387632692279151, 4.99975301202249895294591852784, 6.22343399987458540249644913962, 6.75303557365365576252385644064, 7.44391924969406509446208384299, 8.794063145207066201860999428072, 10.03955547155825308355417878065, 10.83051527936182101389586296933

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