↑

Properties

Label 2-630-315.194-c1-0-30
Degree $2$
Conductor $630$
Sign $0.0951 + 0.995i$
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  + 2-s + (−1.55 − 0.761i)3-s + 4-s + (−1.55 + 1.61i)5-s + (−1.55 − 0.761i)6-s + (−0.405 − 2.61i)7-s + 8-s + (1.83 + 2.36i)9-s + (−1.55 + 1.61i)10-s + (0.922 + 0.532i)11-s + (−1.55 − 0.761i)12-s + (2.83 − 4.91i)13-s + (−0.405 − 2.61i)14-s + (3.63 − 1.32i)15-s + 16-s + (1.18 − 0.686i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.898 − 0.439i)3-s + 0.5·4-s + (−0.693 + 0.720i)5-s + (−0.635 − 0.310i)6-s + (−0.153 − 0.988i)7-s + 0.353·8-s + (0.613 + 0.789i)9-s + (−0.490 + 0.509i)10-s + (0.278 + 0.160i)11-s + (−0.449 − 0.219i)12-s + (0.787 − 1.36i)13-s + (−0.108 − 0.698i)14-s + (0.939 − 0.341i)15-s + 0.250·16-s + (0.288 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0951 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0951 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0951 + 0.995i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.0951 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.989298 - 0.899215i\)
\(L(\frac12)\) \(\approx\) \(0.989298 - 0.899215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.55 + 0.761i)T \)
5 \( 1 + (1.55 - 1.61i)T \)
7 \( 1 + (0.405 + 2.61i)T \)
good11 \( 1 + (-0.922 - 0.532i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.83 + 4.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.18 + 0.686i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.87 + 3.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.24 + 3.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.67 + 2.70i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.52iT - 31T^{2} \)
37 \( 1 + (-7.42 - 4.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.21 + 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.89 - 2.82i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.75iT - 47T^{2} \)
53 \( 1 + (2.29 + 3.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.70T + 59T^{2} \)
61 \( 1 - 9.92iT - 61T^{2} \)
67 \( 1 + 0.0645iT - 67T^{2} \)
71 \( 1 + 6.07iT - 71T^{2} \)
73 \( 1 + (4.78 + 8.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + (7.57 - 4.37i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.74 + 9.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.25 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.64214913092729428730948232233, −10.08603742613282693454300276288, −8.150230906787266764064628010831, −7.59061688608675470300145752129, −6.50614274900927365679794157889, −6.16992449653740584694737327211, −4.67238040828075953526755214250, −3.96374805339087567001698905178, −2.65376535766063391394777872501, −0.68292627367467804434363888037, 1.62488190198444483044500660143, 3.59879182294704408430786575641, 4.29759994174613400227917213464, 5.27228311150534744485919025972, 6.11334764854889866662205106998, 6.85437361144567443300742821249, 8.340824316178905264087406890215, 8.999858120874222455623894981514, 10.06652437674964988236063585575, 11.19934223898539994692487817096

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