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Properties

Label 2-630-315.202-c1-0-12
Degree $2$
Conductor $630$
Sign $0.769 + 0.638i$
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  + (0.258 − 0.965i)2-s + (−1.68 + 0.384i)3-s + (−0.866 − 0.499i)4-s + (−1.94 + 1.10i)5-s + (−0.0656 + 1.73i)6-s + (−2.64 − 0.0747i)7-s + (−0.707 + 0.707i)8-s + (2.70 − 1.29i)9-s + (0.561 + 2.16i)10-s + (0.535 + 0.928i)11-s + (1.65 + 0.511i)12-s + (−1.78 + 0.479i)13-s + (−0.756 + 2.53i)14-s + (2.86 − 2.60i)15-s + (0.500 + 0.866i)16-s + (5.63 − 5.63i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.975 + 0.222i)3-s + (−0.433 − 0.249i)4-s + (−0.870 + 0.492i)5-s + (−0.0267 + 0.706i)6-s + (−0.999 − 0.0282i)7-s + (−0.249 + 0.249i)8-s + (0.901 − 0.432i)9-s + (0.177 + 0.684i)10-s + (0.161 + 0.279i)11-s + (0.477 + 0.147i)12-s + (−0.495 + 0.132i)13-s + (−0.202 + 0.677i)14-s + (0.738 − 0.673i)15-s + (0.125 + 0.216i)16-s + (1.36 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746535 - 0.269565i\)
\(L(\frac12)\) \(\approx\) \(0.746535 - 0.269565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (1.68 - 0.384i)T \)
5 \( 1 + (1.94 - 1.10i)T \)
7 \( 1 + (2.64 + 0.0747i)T \)
good11 \( 1 + (-0.535 - 0.928i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.78 - 0.479i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-5.63 + 5.63i)T - 17iT^{2} \)
19 \( 1 - 4.65T + 19T^{2} \)
23 \( 1 + (-1.07 - 4.01i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.832 + 0.480i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.15 - 2.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.99 + 5.99i)T + 37iT^{2} \)
41 \( 1 + (-4.30 - 2.48i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.88 + 0.505i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-11.8 - 3.16i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.68 + 2.68i)T - 53iT^{2} \)
59 \( 1 + (-4.14 + 7.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.3 - 5.99i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.81 + 0.485i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.780T + 71T^{2} \)
73 \( 1 + (1.59 + 1.59i)T + 73iT^{2} \)
79 \( 1 + (4.01 - 2.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.32 - 4.94i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (-1.36 - 0.364i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.54057515580732215902754330005, −9.845954131121981336989358546300, −9.243826768536568971773914694172, −7.51126259465558650885355898729, −7.03253203427830010311092695530, −5.78308065404654913282849671968, −4.89136255354373981287705953687, −3.75052265015992828674422629498, −2.94501250289521468729342493340, −0.75388785354033343717435814735, 0.825909078346835468445851283444, 3.32045749261086438514796842952, 4.32854006032855064062790671153, 5.40378394479764686340879339686, 6.12050631415094144024911469085, 7.10865136337354690183169227931, 7.80202796508999524319210310729, 8.784848007164994350918191052812, 9.913397344031340311615533928598, 10.60953489607753727129272635521

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