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Properties

Label 2-630-63.47-c1-0-6
Degree $2$
Conductor $630$
Sign $0.0175 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.261 + 1.71i)3-s + (0.499 − 0.866i)4-s + 5-s + (0.629 + 1.61i)6-s + (−2.20 + 1.46i)7-s − 0.999i·8-s + (−2.86 − 0.894i)9-s + (0.866 − 0.5i)10-s + 5.67i·11-s + (1.35 + 1.08i)12-s + (−1.57 + 0.909i)13-s + (−1.17 + 2.37i)14-s + (−0.261 + 1.71i)15-s + (−0.5 − 0.866i)16-s + (2.85 + 4.95i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.150 + 0.988i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.257 + 0.658i)6-s + (−0.832 + 0.553i)7-s − 0.353i·8-s + (−0.954 − 0.298i)9-s + (0.273 − 0.158i)10-s + 1.71i·11-s + (0.390 + 0.312i)12-s + (−0.436 + 0.252i)13-s + (−0.314 + 0.633i)14-s + (−0.0674 + 0.442i)15-s + (−0.125 − 0.216i)16-s + (0.693 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0175 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0175 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0175 - 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.0175 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21657 + 1.19545i\)
\(L(\frac12)\) \(\approx\) \(1.21657 + 1.19545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.261 - 1.71i)T \)
5 \( 1 - T \)
7 \( 1 + (2.20 - 1.46i)T \)
good11 \( 1 - 5.67iT - 11T^{2} \)
13 \( 1 + (1.57 - 0.909i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.85 - 4.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.07 - 0.619i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (2.25 + 1.30i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.72 - 4.45i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.02 - 3.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.21 + 10.7i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.45 - 4.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.02 - 8.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.844 - 0.487i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.68 + 9.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.45 + 3.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0344 + 0.0596i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.62iT - 71T^{2} \)
73 \( 1 + (7.47 - 4.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.683 - 1.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.04 + 10.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.88 + 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.67 - 0.969i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52096723545731230000821824445, −10.02366628066305157972893921364, −9.516266635404038586726509665339, −8.440747939028816047023372500421, −6.93205077339927009414575080513, −6.08144621144429988139316149200, −5.15455480368382392516090258834, −4.32648210681650220043408027995, −3.22499737592056839254439124631, −2.09071287085174235250406916533, 0.77091078306213873114987270448, 2.71040392435146576094190787150, 3.47992652404155194256929096592, 5.22655753238276873169755395212, 5.88458112606880556450563872521, 6.73065359073453720789369626972, 7.48563677939417663581108779705, 8.385929044119331650309995075166, 9.454505367384530127168986817501, 10.49939654864387458728124134393

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