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Properties

Label 2-630-15.2-c3-0-4
Degree $2$
Conductor $630$
Sign $0.458 - 0.888i$
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 + (−11.1 − 0.838i)5-s + (4.94 + 4.94i)7-s + (−5.65 − 5.65i)8-s + (−16.9 + 14.5i)10-s − 62.1i·11-s + (−46.7 + 46.7i)13-s + 14.0·14-s − 16.0·16-s + (−14.9 + 14.9i)17-s − 17.5i·19-s + (−3.35 + 44.5i)20-s + (−87.8 − 87.8i)22-s + (130. + 130. i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.997 − 0.0749i)5-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.536 + 0.461i)10-s − 1.70i·11-s + (−0.997 + 0.997i)13-s + 0.267·14-s − 0.250·16-s + (−0.213 + 0.213i)17-s − 0.212i·19-s + (−0.0374 + 0.498i)20-s + (−0.851 − 0.851i)22-s + (1.18 + 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.458 - 0.888i$
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: \((2,\ 630,\ (\ :3/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.023158628\)
\(L(\frac12)\) \(\approx\) \(1.023158628\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 + 0.838i)T \)
7 \( 1 + (-4.94 - 4.94i)T \)
good11 \( 1 + 62.1iT - 1.33e3T^{2} \)
13 \( 1 + (46.7 - 46.7i)T - 2.19e3iT^{2} \)
17 \( 1 + (14.9 - 14.9i)T - 4.91e3iT^{2} \)
19 \( 1 + 17.5iT - 6.85e3T^{2} \)
23 \( 1 + (-130. - 130. i)T + 1.21e4iT^{2} \)
29 \( 1 - 28.1T + 2.43e4T^{2} \)
31 \( 1 + 276.T + 2.97e4T^{2} \)
37 \( 1 + (-239. - 239. i)T + 5.06e4iT^{2} \)
41 \( 1 + 187. iT - 6.89e4T^{2} \)
43 \( 1 + (219. - 219. i)T - 7.95e4iT^{2} \)
47 \( 1 + (267. - 267. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-434. - 434. i)T + 1.48e5iT^{2} \)
59 \( 1 + 35.1T + 2.05e5T^{2} \)
61 \( 1 + 341.T + 2.26e5T^{2} \)
67 \( 1 + (-526. - 526. i)T + 3.00e5iT^{2} \)
71 \( 1 - 885. iT - 3.57e5T^{2} \)
73 \( 1 + (207. - 207. i)T - 3.89e5iT^{2} \)
79 \( 1 + 150. iT - 4.93e5T^{2} \)
83 \( 1 + (239. + 239. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.63e3T + 7.04e5T^{2} \)
97 \( 1 + (-99.1 - 99.1i)T + 9.12e5iT^{2} \)
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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

−10.71269431418746829107175984419, −9.379084431258759081797109640741, −8.738259923854870348727672788339, −7.70593906004914927711914094905, −6.75658622092537905844759168567, −5.58004274655413327126762146331, −4.68660190725840116890793886678, −3.66685546356074091171567844639, −2.76465975482794387351874096032, −1.17680212065297332470920160378, 0.26986737991625158354199949001, 2.29411866304618361193826665432, 3.55325464712860551365681665672, 4.65185790918439273655585437577, 5.11778961696081378072600390650, 6.73102546493445600644455617804, 7.35843073051831845810493189358, 7.927545985994212195621573907392, 9.038854341654970694982478764869, 10.12033983410837506303911834478

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