↑

Properties

Label 2-630-63.58-c1-0-29
Degree $2$
Conductor $630$
Sign $-0.888 - 0.458i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (1.5 + 0.866i)6-s + (−2 − 1.73i)7-s − 8-s + (1.5 + 2.59i)9-s + (−0.5 − 0.866i)10-s + (2 − 3.46i)11-s + (−1.5 − 0.866i)12-s + (−2 + 3.46i)13-s + (2 + 1.73i)14-s − 1.73i·15-s + 16-s + (−2 − 3.46i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.866 − 0.499i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.158 − 0.273i)10-s + (0.603 − 1.04i)11-s + (−0.433 − 0.249i)12-s + (−0.554 + 0.960i)13-s + (0.534 + 0.462i)14-s − 0.447i·15-s + 0.250·16-s + (−0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 + 9T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.00062353608469862501635957660, −9.497845923632060049865465522062, −8.279515172345285294074953075947, −7.19526906577126156365504043438, −6.62571340683324821487931878187, −5.97084308352458390362534353766, −4.54383236626416990116090455562, −3.10960357396365457400171595436, −1.57410589173899393421346776134, 0, 1.89756787697600957843629335775, 3.50961356314864024846800699743, 4.83220708416201680814919173199, 5.75249642250506811628601377933, 6.60481914812154171274920850709, 7.47284713122457727267891949923, 9.042294803804864084032562136511, 9.183093756932288274538789540247, 10.33512930763482301285433164342

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