↑

Properties

Label 2-630-35.4-c1-0-14
Degree $2$
Conductor $630$
Sign $0.308 + 0.951i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.52 − 1.63i)5-s + (1.13 − 2.38i)7-s + 0.999i·8-s + (−0.499 + 2.17i)10-s + (−0.866 + 1.5i)11-s − 6.50i·13-s + (0.209 + 2.63i)14-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (1.63 + 2.83i)19-s + (−0.656 − 2.13i)20-s − 1.73i·22-s + (−2.83 + 1.63i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.680 − 0.732i)5-s + (0.429 − 0.902i)7-s + 0.353i·8-s + (−0.158 + 0.689i)10-s + (−0.261 + 0.452i)11-s − 1.80i·13-s + (0.0559 + 0.704i)14-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.375 + 0.650i)19-s + (−0.146 − 0.477i)20-s − 0.369i·22-s + (−0.591 + 0.341i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922577 - 0.670470i\)
\(L(\frac12)\) \(\approx\) \(0.922577 - 0.670470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 1.63i)T \)
7 \( 1 + (-1.13 + 2.38i)T \)
good11 \( 1 + (0.866 - 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.50iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.83 - 1.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + (0.137 - 0.238i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.362 - 0.209i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.20T + 41T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 + (-8.03 + 4.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.53 - 3.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.85 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.27 - 9.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.54 - 4.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + (4.54 + 2.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.13 - 3.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.27iT - 83T^{2} \)
89 \( 1 + (4.77 + 8.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.22059877042216361279427877287, −9.679454834251862163076905639997, −8.577358319706622753737410640290, −7.83124722931248117334973761168, −7.12621204611946519582117486959, −5.72311299816141756961067139169, −5.22605060224751604234411747741, −3.86723151190584147124825472035, −2.12719709733582885603816925835, −0.76379159774242715588261558796, 1.85400176168545981182027712626, 2.60065534242842874542529647599, 4.04514008113890946517374245142, 5.45011412685625239019894279481, 6.40246712616847876289534162765, 7.22452489200144593417680965726, 8.376163982030588261341490463888, 9.225323706083552418314952847827, 9.683596566109767424749399407951, 11.01669723850832342120162883516

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