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Properties

Label 2-630-63.58-c1-0-13
Degree $2$
Conductor $630$
Sign $0.208 - 0.977i$
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 + (0.0335 + 1.73i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.0335 + 1.73i)6-s + (2.64 − 0.0963i)7-s + 8-s + (−2.99 + 0.116i)9-s + (0.5 + 0.866i)10-s + (−2.03 + 3.53i)11-s + (0.0335 + 1.73i)12-s + (1.66 − 2.88i)13-s + (2.64 − 0.0963i)14-s + (−1.48 + 0.894i)15-s + 16-s + (2.10 + 3.65i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.0193 + 0.999i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.0137 + 0.706i)6-s + (0.999 − 0.0364i)7-s + 0.353·8-s + (−0.999 + 0.0387i)9-s + (0.158 + 0.273i)10-s + (−0.615 + 1.06i)11-s + (0.00968 + 0.499i)12-s + (0.461 − 0.798i)13-s + (0.706 − 0.0257i)14-s + (−0.382 + 0.231i)15-s + 0.250·16-s + (0.511 + 0.886i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.208 - 0.977i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.208 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92829 + 1.55981i\)
\(L(\frac12)\) \(\approx\) \(1.92829 + 1.55981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0335 - 1.73i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.64 + 0.0963i)T \)
good11 \( 1 + (2.03 - 3.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.66 + 2.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.10 - 3.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.15 + 2.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.365 + 0.632i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.513 - 0.889i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + (-0.647 + 1.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.29 + 5.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.24 - 2.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.50T + 47T^{2} \)
53 \( 1 + (-2.65 - 4.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.96T + 59T^{2} \)
61 \( 1 - 0.903T + 61T^{2} \)
67 \( 1 - 1.10T + 67T^{2} \)
71 \( 1 - 2.91T + 71T^{2} \)
73 \( 1 + (5.73 + 9.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + (6.86 + 11.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.27 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.36 + 12.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.68252367342942221453323559619, −10.29533974954425880464246291806, −9.150665701081766557355932635731, −8.068929998925249670685882763540, −7.28976472315036030934978365420, −5.84437509281888476996242922122, −5.23346119396061951036703673999, −4.31103162310239223192570212254, −3.29847386187783052726129023250, −2.04366575621503336944638919635, 1.22516478384214757262745760357, 2.39947667107946507941848431195, 3.70637927697001606791298192635, 5.18133319391275849453262868295, 5.66884489751127950803831867285, 6.77220117554488531610819245721, 7.76573372427510670519436462144, 8.371905454541027665547833670671, 9.390514717254277652321907681914, 10.87511378289050616784699466820

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