Properties

Label 2-630-5.2-c2-0-12
Degree $2$
Conductor $630$
Sign $-0.779 - 0.626i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + 2i·4-s + (1.66 + 4.71i)5-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s + (−3.04 + 6.38i)10-s + 5.03·11-s + (−2.44 + 2.44i)13-s − 3.74i·14-s − 4·16-s + (18.2 + 18.2i)17-s + 9.56i·19-s + (−9.42 + 3.33i)20-s + (5.03 + 5.03i)22-s + (−16.4 + 16.4i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.333 + 0.942i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.304 + 0.638i)10-s + 0.457·11-s + (−0.188 + 0.188i)13-s − 0.267i·14-s − 0.250·16-s + (1.07 + 1.07i)17-s + 0.503i·19-s + (−0.471 + 0.166i)20-s + (0.228 + 0.228i)22-s + (−0.714 + 0.714i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.779 - 0.626i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ -0.779 - 0.626i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.066587824\)
\(L(\frac12)\) \(\approx\) \(2.066587824\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (-1.66 - 4.71i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 - 5.03T + 121T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 169iT^{2} \)
17 \( 1 + (-18.2 - 18.2i)T + 289iT^{2} \)
19 \( 1 - 9.56iT - 361T^{2} \)
23 \( 1 + (16.4 - 16.4i)T - 529iT^{2} \)
29 \( 1 - 4.18iT - 841T^{2} \)
31 \( 1 + 55.1T + 961T^{2} \)
37 \( 1 + (1.23 + 1.23i)T + 1.36e3iT^{2} \)
41 \( 1 - 12.2T + 1.68e3T^{2} \)
43 \( 1 + (-36.1 + 36.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (18.6 + 18.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (37.8 - 37.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 71.7iT - 3.48e3T^{2} \)
61 \( 1 - 60.8T + 3.72e3T^{2} \)
67 \( 1 + (-30.4 - 30.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 115.T + 5.04e3T^{2} \)
73 \( 1 + (-54.8 + 54.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 + (52.8 - 52.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 16.5iT - 7.92e3T^{2} \)
97 \( 1 + (-71.1 - 71.1i)T + 9.40e3iT^{2} \)
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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

−10.65910053804317251151037032128, −9.956367650562171031354726298443, −8.994810470830259382310553627609, −7.74715839322752722054753947407, −7.17421831218168963206489200259, −6.11996306573819506882197739741, −5.58265676562073295068592391272, −3.99336233735887032856374774512, −3.34983855931482623254423540733, −1.84844874436639353482589924662, 0.63386628930219025604458337573, 2.02218047389505920497110268888, 3.28877486708180313811177421043, 4.48572515806063519232027268655, 5.33721147353663967392351154409, 6.13738321427953553122520376806, 7.36081125301910744261167583877, 8.491347574830020298357695180048, 9.460064294145235792143129991991, 9.845602846036487329565616473448

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