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Properties

Label 2-630-315.212-c1-0-21
Degree $2$
Conductor $630$
Sign $0.961 - 0.276i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (1.06 − 1.36i)3-s − 1.00i·4-s + (1.77 + 1.35i)5-s + (0.210 + 1.71i)6-s + (−2.61 + 0.415i)7-s + (0.707 + 0.707i)8-s + (−0.724 − 2.91i)9-s + (−2.21 + 0.298i)10-s + (3.71 + 2.14i)11-s + (−1.36 − 1.06i)12-s + (0.231 + 0.865i)13-s + (1.55 − 2.14i)14-s + (3.74 − 0.980i)15-s − 1.00·16-s + (2.80 + 0.751i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.615 − 0.787i)3-s − 0.500i·4-s + (0.795 + 0.606i)5-s + (0.0860 + 0.701i)6-s + (−0.987 + 0.157i)7-s + (0.250 + 0.250i)8-s + (−0.241 − 0.970i)9-s + (−0.700 + 0.0945i)10-s + (1.11 + 0.646i)11-s + (−0.393 − 0.307i)12-s + (0.0643 + 0.239i)13-s + (0.415 − 0.572i)14-s + (0.967 − 0.253i)15-s − 0.250·16-s + (0.680 + 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.961 - 0.276i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.961 - 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56700 + 0.220822i\)
\(L(\frac12)\) \(\approx\) \(1.56700 + 0.220822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.06 + 1.36i)T \)
5 \( 1 + (-1.77 - 1.35i)T \)
7 \( 1 + (2.61 - 0.415i)T \)
good11 \( 1 + (-3.71 - 2.14i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.231 - 0.865i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.80 - 0.751i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.19 - 2.99i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.697 + 2.60i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.42 + 5.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.67T + 31T^{2} \)
37 \( 1 + (-2.48 + 0.666i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.97 + 1.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.89 + 1.31i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.365 - 0.365i)T - 47iT^{2} \)
53 \( 1 + (-0.303 + 1.13i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-9.75 - 9.75i)T + 67iT^{2} \)
71 \( 1 + 5.37iT - 71T^{2} \)
73 \( 1 + (13.3 + 3.57i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 + (-1.25 + 4.69i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.56 - 6.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.11 - 11.6i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.07170626934063400698982601988, −9.717478399287558903404555664182, −8.992080561082275392332138086540, −7.900624008028042583129293095537, −6.96110159809469986729210551627, −6.45338969933250979671287717712, −5.66751281652736057142232883741, −3.78139738587707783237067069900, −2.61999131649142641654632424460, −1.35715627899416520641741352208, 1.21343820959907884632681989027, 2.89713123995186807464641935899, 3.58443718569645200267854879325, 4.89076775601529867675639618260, 5.95583156042407783482453953294, 7.14774670372118714684511428417, 8.370775894779636241592743970206, 9.118874779803747876455151775260, 9.640900886311264288900008497427, 10.17105879283846000535319143562

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