↑

Properties

Label 2-630-315.164-c1-0-21
Degree $2$
Conductor $630$
Sign $0.00660 - 0.999i$
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 + (0.241 + 1.71i)3-s + 4-s + (−0.381 + 2.20i)5-s + (0.241 + 1.71i)6-s + (2.63 + 0.178i)7-s + 8-s + (−2.88 + 0.829i)9-s + (−0.381 + 2.20i)10-s + (3.12 − 1.80i)11-s + (0.241 + 1.71i)12-s + (1.68 + 2.92i)13-s + (2.63 + 0.178i)14-s + (−3.87 − 0.121i)15-s + 16-s + (−6.40 − 3.69i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.139 + 0.990i)3-s + 0.5·4-s + (−0.170 + 0.985i)5-s + (0.0987 + 0.700i)6-s + (0.997 + 0.0674i)7-s + 0.353·8-s + (−0.960 + 0.276i)9-s + (−0.120 + 0.696i)10-s + (0.942 − 0.544i)11-s + (0.0698 + 0.495i)12-s + (0.467 + 0.810i)13-s + (0.705 + 0.0477i)14-s + (−0.999 − 0.0314i)15-s + 0.250·16-s + (−1.55 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00660 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00660 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.00660 - 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.00660 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76571 + 1.75409i\)
\(L(\frac12)\) \(\approx\) \(1.76571 + 1.75409i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.241 - 1.71i)T \)
5 \( 1 + (0.381 - 2.20i)T \)
7 \( 1 + (-2.63 - 0.178i)T \)
good11 \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.68 - 2.92i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.40 + 3.69i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.15 - 2.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.99 + 6.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.66 - 4.42i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.44iT - 31T^{2} \)
37 \( 1 + (5.56 - 3.21i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.31 + 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.660 + 0.381i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.22iT - 47T^{2} \)
53 \( 1 + (-1.47 + 2.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 + 1.68iT - 61T^{2} \)
67 \( 1 + 4.74iT - 67T^{2} \)
71 \( 1 - 1.40iT - 71T^{2} \)
73 \( 1 + (-2.78 + 4.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.172T + 79T^{2} \)
83 \( 1 + (-11.5 - 6.65i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.08 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.87812462195847766291968154204, −10.36160513606121298604626183455, −8.793789683718112226168626516310, −8.557932090927332942028307585348, −6.85103518019236842025881086439, −6.43798402800403323103867424721, −4.97202149906944535124098760002, −4.32087722557769144000998894707, −3.34417256388010624577007520123, −2.18140261726275558128729483332, 1.21585693349774985727190965297, 2.22799058519488562945794866599, 3.93700601145118820009081609311, 4.75780085908020170382175362728, 5.85096232211922462220796215532, 6.71475037439555524399212673412, 7.75699520464295786977075948242, 8.468486526972281782168626979845, 9.187240720361841489990449642200, 10.79179428720340362290559486961

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