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Properties

Label 2-630-63.59-c1-0-26
Degree $2$
Conductor $630$
Sign $0.994 + 0.105i$
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  + (0.866 + 0.5i)2-s + (1.13 − 1.30i)3-s + (0.499 + 0.866i)4-s + 5-s + (1.63 − 0.560i)6-s + (2.26 + 1.36i)7-s + 0.999i·8-s + (−0.403 − 2.97i)9-s + (0.866 + 0.5i)10-s − 3.46i·11-s + (1.69 + 0.334i)12-s + (−0.584 − 0.337i)13-s + (1.28 + 2.31i)14-s + (1.13 − 1.30i)15-s + (−0.5 + 0.866i)16-s + (−2.09 + 3.62i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.657 − 0.753i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.669 − 0.228i)6-s + (0.856 + 0.515i)7-s + 0.353i·8-s + (−0.134 − 0.990i)9-s + (0.273 + 0.158i)10-s − 1.04i·11-s + (0.490 + 0.0965i)12-s + (−0.162 − 0.0936i)13-s + (0.342 + 0.618i)14-s + (0.294 − 0.336i)15-s + (−0.125 + 0.216i)16-s + (−0.507 + 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.89665 - 0.152855i\)
\(L(\frac12)\) \(\approx\) \(2.89665 - 0.152855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.13 + 1.30i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.26 - 1.36i)T \)
good11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (0.584 + 0.337i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.09 - 3.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.18 + 0.683i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.10iT - 23T^{2} \)
29 \( 1 + (4.77 - 2.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 - 0.724i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.59 - 7.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.79 + 8.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.38 + 2.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.50 + 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.77 + 3.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.15 + 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.15 + 2.97i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.90 + 3.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 + (-2.54 - 1.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.65 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.69 + 2.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.11 + 5.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.750 - 0.433i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.84651811084952600489697210946, −9.404019414510098753135847972098, −8.578008194342694605387701831706, −7.996594959654121877284011530792, −6.98343485094473986201787662515, −6.02003703502116096579612072447, −5.30348695964306263131201697293, −3.88280073879582782237845002930, −2.75096740448407938603786580239, −1.62109101569272911962749944138, 1.81747733663075277896345412573, 2.81573085965642035787841271506, 4.28963872195869567476454162214, 4.65818170815820632703554668090, 5.77142834294397259470767541177, 7.20267991789876238165876758151, 7.894902249374616429247548400699, 9.269640536685507194506576976761, 9.674841099697811535178294763567, 10.72355526965529448760920080003

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