↑

Properties

Label 2-630-21.17-c1-0-6
Degree $2$
Conductor $630$
Sign $-0.958 + 0.286i$
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 + (−0.5 − 0.866i)5-s + (−0.189 − 2.63i)7-s + 0.999i·8-s + (0.866 + 0.499i)10-s + (−3.44 − 1.99i)11-s + 0.0681i·13-s + (1.48 + 2.19i)14-s + (−0.5 − 0.866i)16-s + (−3.66 + 6.34i)17-s + (−1.76 + 1.01i)19-s − 0.999·20-s + 3.98·22-s + (−3.23 + 1.86i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.0716 − 0.997i)7-s + 0.353i·8-s + (0.273 + 0.158i)10-s + (−1.03 − 0.600i)11-s + 0.0189i·13-s + (0.396 + 0.585i)14-s + (−0.125 − 0.216i)16-s + (−0.888 + 1.53i)17-s + (−0.404 + 0.233i)19-s − 0.223·20-s + 0.848·22-s + (−0.673 + 0.389i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.958 + 0.286i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.958 + 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0282796 - 0.193456i\)
\(L(\frac12)\) \(\approx\) \(0.0282796 - 0.193456i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.189 + 2.63i)T \)
good11 \( 1 + (3.44 + 1.99i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0681iT - 13T^{2} \)
17 \( 1 + (3.66 - 6.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.76 - 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.23 - 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.03 - 3.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 + 0.964T + 43T^{2} \)
47 \( 1 + (-0.830 - 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.4 + 6.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.32 + 9.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 + 3.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.33 - 9.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.93iT - 71T^{2} \)
73 \( 1 + (10.0 + 5.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.77 + 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + (-0.913 - 1.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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

−10.31546504846815969580290293371, −9.275862001262540501904592605619, −8.174757531066787545904381922711, −7.895297576267191526320637519069, −6.67920008947397773452760665914, −5.84152771327479995139784947801, −4.62677568209032346894125995635, −3.56118003822971762037807863508, −1.82110237693977513153186222654, −0.12126170927981592191683674212, 2.20925115754573028705030751703, 2.90145497392886344799047429300, 4.47007198122932585161785602628, 5.54345663086696495851731013245, 6.77520095482170101748871156252, 7.53844088843443022076354448678, 8.521191466312951678994839332579, 9.266380633169907922894416959656, 10.10080999047280019349851208290, 10.96908409577019558511002513609

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