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Properties

Label 2-630-5.3-c2-0-26
Degree $2$
Conductor $630$
Sign $-0.963 - 0.267i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s − 2i·4-s + (−4.80 + 1.39i)5-s + (1.87 − 1.87i)7-s + (−2 − 2i)8-s + (−3.40 + 6.19i)10-s + 11.5·11-s + (−7.70 − 7.70i)13-s − 3.74i·14-s − 4·16-s + (−21.0 + 21.0i)17-s − 24.1i·19-s + (2.78 + 9.60i)20-s + (11.5 − 11.5i)22-s + (−30.1 − 30.1i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (−0.960 + 0.278i)5-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.340 + 0.619i)10-s + 1.04·11-s + (−0.592 − 0.592i)13-s − 0.267i·14-s − 0.250·16-s + (−1.23 + 1.23i)17-s − 1.26i·19-s + (0.139 + 0.480i)20-s + (0.523 − 0.523i)22-s + (−1.31 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.963 - 0.267i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ -0.963 - 0.267i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5335413149\)
\(L(\frac12)\) \(\approx\) \(0.5335413149\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
5 \( 1 + (4.80 - 1.39i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 - 11.5T + 121T^{2} \)
13 \( 1 + (7.70 + 7.70i)T + 169iT^{2} \)
17 \( 1 + (21.0 - 21.0i)T - 289iT^{2} \)
19 \( 1 + 24.1iT - 361T^{2} \)
23 \( 1 + (30.1 + 30.1i)T + 529iT^{2} \)
29 \( 1 - 51.1iT - 841T^{2} \)
31 \( 1 + 46.9T + 961T^{2} \)
37 \( 1 + (8.50 - 8.50i)T - 1.36e3iT^{2} \)
41 \( 1 + 18.6T + 1.68e3T^{2} \)
43 \( 1 + (26.1 + 26.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (50.1 - 50.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (7.08 + 7.08i)T + 2.80e3iT^{2} \)
59 \( 1 + 94.3iT - 3.48e3T^{2} \)
61 \( 1 - 8.09T + 3.72e3T^{2} \)
67 \( 1 + (20.6 - 20.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 63.7T + 5.04e3T^{2} \)
73 \( 1 + (-50.1 - 50.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 1.06iT - 6.24e3T^{2} \)
83 \( 1 + (53.6 + 53.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-23.7 + 23.7i)T - 9.40e3iT^{2} \)
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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

−10.24800824748105252771723035643, −8.988097930728784376535671534247, −8.278418182768578422682018693230, −7.04384361915275524173084432617, −6.44234128970311062603288649833, −4.95380623738316995936411631167, −4.17438106307174210020591956434, −3.29594449009617955060162955071, −1.88841872466207016169946064530, −0.15772915920875024498805156375, 1.96586391306389255981984463901, 3.65064430224004530732477689352, 4.29896940390841311117373392191, 5.31832221263808199749223832537, 6.43760261312359825042558612250, 7.33894880531124377836642294703, 8.049077499944193984678908045777, 9.027561140420024271919894736930, 9.754784252796543550017522334955, 11.36095352114511799789600085862

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