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Properties

Label 2-630-35.9-c1-0-18
Degree $2$
Conductor $630$
Sign $-0.951 + 0.308i$
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 + (0.499 + 0.866i)4-s + (−1.23 − 1.86i)5-s + (0.866 − 2.5i)7-s − 0.999i·8-s + (0.133 + 2.23i)10-s − 2i·13-s + (−2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (−1 + 1.73i)19-s + (1 − 1.99i)20-s + (−0.866 − 0.5i)23-s + (−1.96 + 4.59i)25-s + (−1 + 1.73i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.550 − 0.834i)5-s + (0.327 − 0.944i)7-s − 0.353i·8-s + (0.0423 + 0.705i)10-s − 0.554i·13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.229 + 0.397i)19-s + (0.223 − 0.447i)20-s + (−0.180 − 0.104i)23-s + (−0.392 + 0.919i)25-s + (−0.196 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.951 + 0.308i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0985095 - 0.623391i\)
\(L(\frac12)\) \(\approx\) \(0.0985095 - 0.623391i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.23 + 1.86i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
good11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.20959440450676560102535521776, −9.360930021394684267099668046165, −8.365317777142687758103929442774, −7.82993786932585425364050054859, −6.94105872547271222209475223437, −5.56036853135651285061319521273, −4.35851963907052950862976603690, −3.58331237083444002751491872950, −1.82404244475852572190672760076, −0.41265672949778514037274026401, 1.95888051205774662574856391791, 3.16153830379898705716468259628, 4.61039552025307951790727408926, 5.74369347861119535708460193043, 6.75681298426861279425124809933, 7.38571479891261352272831902850, 8.516294120510316841543845411563, 9.003180390594203234399275612652, 10.13301459841786638068726994709, 10.94756842002999471335904082259

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