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Properties

Label 2-630-15.2-c3-0-17
Degree $2$
Conductor $630$
Sign $0.943 - 0.329i$
Analytic cond. $37.1712$
Root an. cond. $6.09681$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)2-s − 4.00i·4-s + (−10.3 + 4.33i)5-s + (4.94 + 4.94i)7-s + (5.65 + 5.65i)8-s + (8.44 − 20.7i)10-s − 8.72i·11-s + (3.39 − 3.39i)13-s − 14.0·14-s − 16.0·16-s + (−35.6 + 35.6i)17-s − 87.4i·19-s + (17.3 + 41.2i)20-s + (12.3 + 12.3i)22-s + (−8.19 − 8.19i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.921 + 0.387i)5-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.267 − 0.654i)10-s − 0.239i·11-s + (0.0724 − 0.0724i)13-s − 0.267·14-s − 0.250·16-s + (−0.508 + 0.508i)17-s − 1.05i·19-s + (0.193 + 0.460i)20-s + (0.119 + 0.119i)22-s + (−0.0743 − 0.0743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.329i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.943 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.943 - 0.329i$
Analytic conductor: \(37.1712\)
Root analytic conductor: \(6.09681\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :3/2),\ 0.943 - 0.329i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.023491240\)
\(L(\frac12)\) \(\approx\) \(1.023491240\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 1.41i)T \)
3 \( 1 \)
5 \( 1 + (10.3 - 4.33i)T \)
7 \( 1 + (-4.94 - 4.94i)T \)
good11 \( 1 + 8.72iT - 1.33e3T^{2} \)
13 \( 1 + (-3.39 + 3.39i)T - 2.19e3iT^{2} \)
17 \( 1 + (35.6 - 35.6i)T - 4.91e3iT^{2} \)
19 \( 1 + 87.4iT - 6.85e3T^{2} \)
23 \( 1 + (8.19 + 8.19i)T + 1.21e4iT^{2} \)
29 \( 1 + 199.T + 2.43e4T^{2} \)
31 \( 1 + 21.6T + 2.97e4T^{2} \)
37 \( 1 + (-6.63 - 6.63i)T + 5.06e4iT^{2} \)
41 \( 1 - 95.4iT - 6.89e4T^{2} \)
43 \( 1 + (-144. + 144. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-30.7 + 30.7i)T - 1.03e5iT^{2} \)
53 \( 1 + (-221. - 221. i)T + 1.48e5iT^{2} \)
59 \( 1 - 531.T + 2.05e5T^{2} \)
61 \( 1 - 578.T + 2.26e5T^{2} \)
67 \( 1 + (-222. - 222. i)T + 3.00e5iT^{2} \)
71 \( 1 + 134. iT - 3.57e5T^{2} \)
73 \( 1 + (-111. + 111. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.23e3iT - 4.93e5T^{2} \)
83 \( 1 + (-336. - 336. i)T + 5.71e5iT^{2} \)
89 \( 1 - 509.T + 7.04e5T^{2} \)
97 \( 1 + (-408. - 408. i)T + 9.12e5iT^{2} \)
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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

−10.28113301201416396577629763657, −9.099830805426542172251693750295, −8.491760111781366537566174424597, −7.58632926104814931003367874539, −6.88412095771203095684512733119, −5.86648161581925251700144462298, −4.74232824419703786093163969366, −3.65428122616987019791839751397, −2.26615890543214910229792300003, −0.56672771322763028923409433159, 0.71420673217140635305481891297, 2.03884604510448222768970369539, 3.53480297453383336763306805469, 4.28623338792144656192041694261, 5.43182488335245400985163980621, 6.91593368509429567592984612920, 7.68268795014470296864304466564, 8.418929445375514823264728874810, 9.274012658655828097823452338554, 10.15726688517056861065334902044

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