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Properties

Label 2-630-63.38-c1-0-22
Degree $2$
Conductor $630$
Sign $0.948 - 0.317i$
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  + i·2-s + (−0.560 + 1.63i)3-s − 4-s + (0.5 − 0.866i)5-s + (−1.63 − 0.560i)6-s + (0.0477 − 2.64i)7-s − i·8-s + (−2.37 − 1.83i)9-s + (0.866 + 0.5i)10-s + (3.00 − 1.73i)11-s + (0.560 − 1.63i)12-s + (0.584 − 0.337i)13-s + (2.64 + 0.0477i)14-s + (1.13 + 1.30i)15-s + 16-s + (2.09 − 3.62i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.323 + 0.946i)3-s − 0.5·4-s + (0.223 − 0.387i)5-s + (−0.669 − 0.228i)6-s + (0.0180 − 0.999i)7-s − 0.353i·8-s + (−0.790 − 0.611i)9-s + (0.273 + 0.158i)10-s + (0.905 − 0.522i)11-s + (0.161 − 0.473i)12-s + (0.162 − 0.0936i)13-s + (0.706 + 0.0127i)14-s + (0.294 + 0.336i)15-s + 0.250·16-s + (0.507 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.948 - 0.317i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.948 - 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25201 + 0.203863i\)
\(L(\frac12)\) \(\approx\) \(1.25201 + 0.203863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.560 - 1.63i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.0477 + 2.64i)T \)
good11 \( 1 + (-3.00 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.584 + 0.337i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.09 + 3.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.18 + 0.683i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.81 - 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.77 + 2.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.44iT - 31T^{2} \)
37 \( 1 + (-4.59 - 7.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.79 + 8.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.38 - 2.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.00T + 47T^{2} \)
53 \( 1 + (-6.77 - 3.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.31T + 59T^{2} \)
61 \( 1 - 5.95iT - 61T^{2} \)
67 \( 1 - 3.80T + 67T^{2} \)
71 \( 1 + 15.0iT - 71T^{2} \)
73 \( 1 + (-2.54 - 1.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + (-1.69 + 2.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.11 - 5.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.750 - 0.433i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.49783776969984396001542715131, −9.594200019395014946965407665507, −9.092306918775365744356624615608, −8.024591733252341433458217127081, −6.99346490577566561359568903680, −6.03478163231426989695539913032, −5.15241455782973283528001654301, −4.23304489286829115120389416265, −3.36169495778805265473095965820, −0.818991834089344501216762285365, 1.46125272823907028214407264207, 2.40830818029516888438725642037, 3.66666856432807013718288634969, 5.18968054036664049703467066106, 6.02685728168894812008800017380, 6.89898666788585388975570223371, 7.998201047498700912660173315215, 8.894789505670922306240775360327, 9.691257298789103769253616311859, 10.80410165273602983780538721946

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