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Properties

Label 2-630-105.89-c1-0-4
Degree $2$
Conductor $630$
Sign $0.269 - 0.962i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.11 − 0.712i)5-s + (1.63 + 2.07i)7-s + 0.999·8-s + (−0.442 + 2.19i)10-s + (−5.48 + 3.16i)11-s + 1.05·13-s + (−2.61 + 0.381i)14-s + (−0.5 + 0.866i)16-s + (4.01 − 2.31i)17-s + (5.35 + 3.08i)19-s + (−1.67 − 1.47i)20-s − 6.33i·22-s + (−3.52 + 6.10i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.947 − 0.318i)5-s + (0.619 + 0.784i)7-s + 0.353·8-s + (−0.139 + 0.693i)10-s + (−1.65 + 0.954i)11-s + 0.292·13-s + (−0.699 + 0.101i)14-s + (−0.125 + 0.216i)16-s + (0.973 − 0.562i)17-s + (1.22 + 0.708i)19-s + (−0.374 − 0.330i)20-s − 1.34i·22-s + (−0.734 + 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.269 - 0.962i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.269 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14388 + 0.867502i\)
\(L(\frac12)\) \(\approx\) \(1.14388 + 0.867502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.11 + 0.712i)T \)
7 \( 1 + (-1.63 - 2.07i)T \)
good11 \( 1 + (5.48 - 3.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
17 \( 1 + (-4.01 + 2.31i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.35 - 3.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.52 - 6.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.98iT - 29T^{2} \)
31 \( 1 + (-5.46 + 3.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.01 - 1.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.97T + 41T^{2} \)
43 \( 1 - 2.58iT - 43T^{2} \)
47 \( 1 + (-7.06 - 4.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.43 - 7.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.452 - 0.783i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.81 + 5.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.51 - 4.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (-4.18 - 7.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.73 + 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + (-1.91 + 3.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.87T + 97T^{2} \)
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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

−10.34513618834083236273458743291, −9.848714972993016452084348812128, −9.117179843740139225908161959908, −7.83227875576834761301924738774, −7.67759383749798636091943190500, −6.00893571576065106605634651564, −5.47132674777541729462603628894, −4.74304333174391314021438652953, −2.76451248598314408872864353796, −1.54584866137885965941625113266, 0.991298247239201948069731159759, 2.46538099441130436783996405910, 3.42883153058017667942650879946, 4.92943617031749326436650936681, 5.71941533359616529332726331696, 7.00402879768649284457673901600, 7.996885009796333476227101528189, 8.624374321774532393812225845285, 9.909390326654328696814630913750, 10.47092591919244511914248289008

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