↑

Properties

Label 2-630-63.47-c1-0-17
Degree $2$
Conductor $630$
Sign $-0.592 + 0.805i$
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 + (−0.866 − 1.5i)3-s + (0.499 − 0.866i)4-s − 5-s + (1.5 + 0.866i)6-s + (0.5 + 2.59i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (0.866 − 0.5i)10-s − 4.73i·11-s − 1.73·12-s + (3 − 1.73i)13-s + (−1.73 − 2i)14-s + (0.866 + 1.5i)15-s + (−0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.612 + 0.353i)6-s + (0.188 + 0.981i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.273 − 0.158i)10-s − 1.42i·11-s − 0.499·12-s + (0.832 − 0.480i)13-s + (−0.462 − 0.534i)14-s + (0.223 + 0.387i)15-s + (−0.125 − 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.223224 - 0.441612i\)
\(L(\frac12)\) \(\approx\) \(0.223224 - 0.441612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 + 4.73iT - 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.53iT - 23T^{2} \)
29 \( 1 + (5.59 + 3.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.09 + 4.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.09 - 5.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.59 - 2.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.29 - 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.09 + 1.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.1 + 6.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.73iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.29 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.59 + 9.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.19 - 3.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.39 - 4.26i)T + (48.5 + 84.0i)T^{2} \)
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

−10.49317487745292279000599243155, −9.050583950586621629954900932299, −8.393240649343308236261571448313, −7.85306146994668876273652100038, −6.65907050825912759500501845490, −5.90931070942197429662999659403, −5.23725999504448322484085722535, −3.37429495985151954643688675634, −1.94552673553787370584648428394, −0.35889196522443064424264991280, 1.56261958949668680484042128166, 3.52683880503321538642398586293, 4.17227448173198733089592572134, 5.21127362825560431148331972097, 6.68928564122393109300222088875, 7.35421664463322503674973931848, 8.461937319155074109746802266127, 9.426327412580283132929633999148, 10.03497049684643941034137921375, 10.97235769134560927274016142990

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