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Properties

Label 2-63-21.5-c5-0-0
Degree $2$
Conductor $63$
Sign $-0.986 + 0.164i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.49 − 5.47i)2-s + (44.0 + 76.2i)4-s + (−41.3 + 71.5i)5-s + (−18.0 + 128. i)7-s − 614. i·8-s + (784. − 452. i)10-s + (−98.5 + 56.9i)11-s − 329. i·13-s + (875. − 1.11e3i)14-s + (−1.95e3 + 3.39e3i)16-s + (221. + 383. i)17-s + (−1.45e3 − 839. i)19-s − 7.27e3·20-s + 1.24e3·22-s + (194. + 112. i)23-s + ⋯
L(s)  = 1  + (−1.67 − 0.968i)2-s + (1.37 + 2.38i)4-s + (−0.738 + 1.27i)5-s + (−0.139 + 0.990i)7-s − 3.39i·8-s + (2.47 − 1.43i)10-s + (−0.245 + 0.141i)11-s − 0.540i·13-s + (1.19 − 1.52i)14-s + (−1.91 + 3.31i)16-s + (0.185 + 0.322i)17-s + (−0.924 − 0.533i)19-s − 4.06·20-s + 0.549·22-s + (0.0764 + 0.0441i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.986 + 0.164i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.986 + 0.164i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00180198 - 0.0218058i\)
\(L(\frac12)\) \(\approx\) \(0.00180198 - 0.0218058i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (18.0 - 128. i)T \)
good2 \( 1 + (9.49 + 5.47i)T + (16 + 27.7i)T^{2} \)
5 \( 1 + (41.3 - 71.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (98.5 - 56.9i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 329. iT - 3.71e5T^{2} \)
17 \( 1 + (-221. - 383. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.45e3 + 839. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-194. - 112. i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.98e3iT - 2.05e7T^{2} \)
31 \( 1 + (-2.92e3 + 1.68e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-5.85e3 + 1.01e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 1.05e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.10e4 + 1.91e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (3.94e3 - 2.27e3i)T + (2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (6.66e3 + 1.15e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (4.39e4 + 2.53e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.56e4 - 4.44e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 1.20e3iT - 1.80e9T^{2} \)
73 \( 1 + (-7.65e3 + 4.41e3i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.17e4 - 3.77e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 5.81e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.42e4 + 9.39e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 5.44e3iT - 8.58e9T^{2} \)
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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

−15.04005509207597377456268974974, −12.81935329087339156908465065874, −11.82519506079050739561631573754, −10.95137879013917227791215337560, −10.14249486937257349280546892278, −8.809364976230705754995937151266, −7.81139035645894039932215507250, −6.65920114901746177853890564621, −3.38601271718089133325969490960, −2.31510303499890863511676799992, 0.01935928710568571272698596700, 1.23144596210316946263302826217, 4.70549527714029391294765669372, 6.42744977766794756009132126631, 7.72726601085278865081567841845, 8.441435119561217670117805761576, 9.562630806360850076788198000758, 10.65459231876923843795355144383, 11.92008042116768886819066974355, 13.63453052239020306161858545243

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