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Properties

Label 2-630-63.59-c1-0-16
Degree $2$
Conductor $630$
Sign $0.0497 - 0.998i$
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 + (1.46 + 0.921i)3-s + (0.499 + 0.866i)4-s + 5-s + (0.809 + 1.53i)6-s + (−1.89 + 1.84i)7-s + 0.999i·8-s + (1.30 + 2.70i)9-s + (0.866 + 0.5i)10-s − 0.645i·11-s + (−0.0648 + 1.73i)12-s + (−0.230 − 0.133i)13-s + (−2.56 + 0.647i)14-s + (1.46 + 0.921i)15-s + (−0.5 + 0.866i)16-s + (−0.525 + 0.910i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.846 + 0.532i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.330 + 0.625i)6-s + (−0.717 + 0.696i)7-s + 0.353i·8-s + (0.433 + 0.901i)9-s + (0.273 + 0.158i)10-s − 0.194i·11-s + (−0.0187 + 0.499i)12-s + (−0.0639 − 0.0369i)13-s + (−0.685 + 0.173i)14-s + (0.378 + 0.237i)15-s + (−0.125 + 0.216i)16-s + (−0.127 + 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0497 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0497 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0497 - 0.998i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.0497 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98258 + 1.88626i\)
\(L(\frac12)\) \(\approx\) \(1.98258 + 1.88626i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 - 0.921i)T \)
5 \( 1 - T \)
7 \( 1 + (1.89 - 1.84i)T \)
good11 \( 1 + 0.645iT - 11T^{2} \)
13 \( 1 + (0.230 + 0.133i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.525 - 0.910i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.938 + 0.541i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + (-2.95 + 1.70i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.57 + 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.37 - 2.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.16 + 3.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.04 + 8.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0268 - 0.0465i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.1 - 5.86i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.85 + 4.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.06 - 1.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.61 + 6.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (13.9 + 8.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.16 + 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.83 + 3.16i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.8 - 8.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.60265109567524798764794568267, −9.879726779266230184071127057526, −8.949932155107184181304848460351, −8.355270778811826833398892854534, −7.18460387041655648711486374212, −6.20786073531238858755563765613, −5.29730877717881219007037377380, −4.23347917824521063455025138981, −3.12737429473217296306787935215, −2.28139553899454484745299676501, 1.26235271461507935641833174500, 2.63261406444890443465657146219, 3.50995593060315459274666774542, 4.58819457863519671623336807730, 5.96596434283389623743333541273, 6.81618316566063794092544887938, 7.54961215976087250709778143042, 8.740254365184185180179844840814, 9.718922676982878797827800488610, 10.16525269702762849159577418829

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