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Properties

Label 2-630-63.25-c1-0-21
Degree $2$
Conductor $630$
Sign $0.919 - 0.393i$
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 + (1.31 + 1.12i)3-s + 4-s + (0.5 − 0.866i)5-s + (1.31 + 1.12i)6-s + (2.64 + 0.0963i)7-s + 8-s + (0.465 + 2.96i)9-s + (0.5 − 0.866i)10-s + (−1.10 − 1.91i)11-s + (1.31 + 1.12i)12-s + (−0.313 − 0.542i)13-s + (2.64 + 0.0963i)14-s + (1.63 − 0.576i)15-s + 16-s + (−1.90 + 3.29i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.759 + 0.649i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.537 + 0.459i)6-s + (0.999 + 0.0364i)7-s + 0.353·8-s + (0.155 + 0.987i)9-s + (0.158 − 0.273i)10-s + (−0.332 − 0.576i)11-s + (0.379 + 0.324i)12-s + (−0.0868 − 0.150i)13-s + (0.706 + 0.0257i)14-s + (0.421 − 0.148i)15-s + 0.250·16-s + (−0.461 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.919 - 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06032 + 0.628159i\)
\(L(\frac12)\) \(\approx\) \(3.06032 + 0.628159i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 - 1.12i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.64 - 0.0963i)T \)
good11 \( 1 + (1.10 + 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.313 + 0.542i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.90 - 3.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.79 + 6.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.30 - 2.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 - 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.40T + 31T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.09 + 1.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.25 - 7.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.450T + 47T^{2} \)
53 \( 1 + (1.19 - 2.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.80T + 59T^{2} \)
61 \( 1 + 0.968T + 61T^{2} \)
67 \( 1 - 4.08T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 + (-0.696 + 1.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (7.35 - 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.98 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.36 + 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.88468153055124058832132654063, −9.829159636191226927730208562562, −8.676939058232671551208457829427, −8.285748858266991753971905668535, −7.15280502755105676481527341895, −5.83504158075142378682562686146, −4.87075941769679926317220385255, −4.26425168227647019278900165772, −2.97123558910428624665446946636, −1.86420157251388859054062571516, 1.75224994645616744017924673477, 2.56536800905053654001528566030, 3.90855885079563034821945847872, 4.88347104654788625421699683100, 6.14541712149112601846736828972, 6.95706212970612733020793815495, 7.87028169418165416903095861671, 8.484972123485659648539486498887, 9.785306878084190235425008582002, 10.55291750381440447410365807089

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