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Properties

Label 2-630-105.59-c1-0-6
Degree $2$
Conductor $630$
Sign $0.673 + 0.739i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.442 − 2.19i)5-s + (−1.63 + 2.07i)7-s + 0.999·8-s + (−2.11 + 0.712i)10-s + (5.48 + 3.16i)11-s − 1.05·13-s + (2.61 + 0.381i)14-s + (−0.5 − 0.866i)16-s + (4.01 + 2.31i)17-s + (5.35 − 3.08i)19-s + (1.67 + 1.47i)20-s − 6.33i·22-s + (−3.52 − 6.10i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.197 − 0.980i)5-s + (−0.619 + 0.784i)7-s + 0.353·8-s + (−0.670 + 0.225i)10-s + (1.65 + 0.954i)11-s − 0.292·13-s + (0.699 + 0.101i)14-s + (−0.125 − 0.216i)16-s + (0.973 + 0.562i)17-s + (1.22 − 0.708i)19-s + (0.374 + 0.330i)20-s − 1.34i·22-s + (−0.734 − 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18455 - 0.523695i\)
\(L(\frac12)\) \(\approx\) \(1.18455 - 0.523695i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.442 + 2.19i)T \)
7 \( 1 + (1.63 - 2.07i)T \)
good11 \( 1 + (-5.48 - 3.16i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.05T + 13T^{2} \)
17 \( 1 + (-4.01 - 2.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.35 + 3.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.52 + 6.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.98iT - 29T^{2} \)
31 \( 1 + (-5.46 - 3.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.01 - 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.97T + 41T^{2} \)
43 \( 1 - 2.58iT - 43T^{2} \)
47 \( 1 + (-7.06 + 4.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.43 + 7.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.452 - 0.783i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.81 - 5.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.51 - 4.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (4.18 - 7.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.73 - 4.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + (1.91 + 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.87T + 97T^{2} \)
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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

−10.13098647480270088255192020601, −9.611578263745906800538499368254, −8.987508899966151595037200724303, −8.171864247036336139450069510431, −6.93973894921289879191605382083, −5.91986173726522860581677508158, −4.77482093936656810843290898246, −3.80579333299252163590139245416, −2.39933427025653360372973166098, −1.10427618754649080259582882028, 1.15264410315692589552192223400, 3.20120640839776694800145378496, 3.94144809653768400870460386340, 5.68229847021605599483985648194, 6.26681065057505184234499846381, 7.26068154094580248494007457736, 7.73404001834427282058410863664, 9.216403574167254837061273290686, 9.698645090029175640441931594556, 10.49215941952518536745481660787

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