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Properties

Label 2-630-15.2-c3-0-23
Degree $2$
Conductor $630$
Sign $0.0947 + 0.995i$
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 + (1.74 + 11.0i)5-s + (−4.94 − 4.94i)7-s + (5.65 + 5.65i)8-s + (−18.0 − 13.1i)10-s + 2.28i·11-s + (25.7 − 25.7i)13-s + 14.0·14-s − 16.0·16-s + (−71.5 + 71.5i)17-s − 50.6i·19-s + (44.1 − 6.98i)20-s + (−3.23 − 3.23i)22-s + (−53.7 − 53.7i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.156 + 0.987i)5-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.571 − 0.415i)10-s + 0.0627i·11-s + (0.549 − 0.549i)13-s + 0.267·14-s − 0.250·16-s + (−1.02 + 1.02i)17-s − 0.611i·19-s + (0.493 − 0.0780i)20-s + (−0.0313 − 0.0313i)22-s + (−0.487 − 0.487i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0947 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0947 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0947 + 0.995i$
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: \((2,\ 630,\ (\ :3/2),\ 0.0947 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4643549060\)
\(L(\frac12)\) \(\approx\) \(0.4643549060\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.74 - 11.0i)T \)
7 \( 1 + (4.94 + 4.94i)T \)
good11 \( 1 - 2.28iT - 1.33e3T^{2} \)
13 \( 1 + (-25.7 + 25.7i)T - 2.19e3iT^{2} \)
17 \( 1 + (71.5 - 71.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 50.6iT - 6.85e3T^{2} \)
23 \( 1 + (53.7 + 53.7i)T + 1.21e4iT^{2} \)
29 \( 1 - 186.T + 2.43e4T^{2} \)
31 \( 1 + 211.T + 2.97e4T^{2} \)
37 \( 1 + (70.1 + 70.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 50.2iT - 6.89e4T^{2} \)
43 \( 1 + (102. - 102. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-216. + 216. i)T - 1.03e5iT^{2} \)
53 \( 1 + (91.3 + 91.3i)T + 1.48e5iT^{2} \)
59 \( 1 + 234.T + 2.05e5T^{2} \)
61 \( 1 + 22.8T + 2.26e5T^{2} \)
67 \( 1 + (267. + 267. i)T + 3.00e5iT^{2} \)
71 \( 1 + 408. iT - 3.57e5T^{2} \)
73 \( 1 + (-15.6 + 15.6i)T - 3.89e5iT^{2} \)
79 \( 1 + 426. iT - 4.93e5T^{2} \)
83 \( 1 + (659. + 659. i)T + 5.71e5iT^{2} \)
89 \( 1 - 331.T + 7.04e5T^{2} \)
97 \( 1 + (875. + 875. i)T + 9.12e5iT^{2} \)
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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

−10.12172176439441190338755308060, −9.031968330032814626090029480245, −8.232476321651897297252485468079, −7.22993630211233663035638046070, −6.50271297928474964528619462358, −5.80627231864993766567571035175, −4.38163427750618067184688690136, −3.18077789888685880537899688313, −1.88779999494582324714239831690, −0.16526238789031400076098982499, 1.21120758590675323706953763806, 2.34332134572628696854141307365, 3.74434119572147679118435840124, 4.74375301065508561799683580929, 5.85092062035341653980544618605, 6.93570224175945804132488563914, 8.056498816326432643178856872308, 8.899692771847225042811626970595, 9.356801291102008581271066950192, 10.29148584329868389919913922681

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