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Properties

Label 2-63-7.4-c5-0-12
Degree $2$
Conductor $63$
Sign $0.919 + 0.391i$
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  + (3.54 + 6.13i)2-s + (−9.08 + 15.7i)4-s + (−39.9 − 69.1i)5-s + (43.1 − 122. i)7-s + 97.9·8-s + (282. − 489. i)10-s + (175. − 304. i)11-s − 291.·13-s + (902. − 168. i)14-s + (637. + 1.10e3i)16-s + (−185. + 320. i)17-s + (−752. − 1.30e3i)19-s + 1.45e3·20-s + 2.49e3·22-s + (−212. − 368. i)23-s + ⋯
L(s)  = 1  + (0.626 + 1.08i)2-s + (−0.283 + 0.491i)4-s + (−0.713 − 1.23i)5-s + (0.332 − 0.942i)7-s + 0.541·8-s + (0.893 − 1.54i)10-s + (0.438 − 0.759i)11-s − 0.478·13-s + (1.23 − 0.229i)14-s + (0.622 + 1.07i)16-s + (−0.155 + 0.268i)17-s + (−0.478 − 0.828i)19-s + 0.810·20-s + 1.09·22-s + (−0.0839 − 0.145i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.391i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.919 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.919 + 0.391i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.919 + 0.391i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.08878 - 0.426420i\)
\(L(\frac12)\) \(\approx\) \(2.08878 - 0.426420i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-43.1 + 122. i)T \)
good2 \( 1 + (-3.54 - 6.13i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (39.9 + 69.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-175. + 304. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 291.T + 3.71e5T^{2} \)
17 \( 1 + (185. - 320. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (752. + 1.30e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (212. + 368. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 7.78e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.28e3 + 2.23e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (369. + 640. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 7.02e3T + 1.15e8T^{2} \)
43 \( 1 - 1.83e3T + 1.47e8T^{2} \)
47 \( 1 + (766. + 1.32e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.76e3 - 8.25e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.48e4 - 2.56e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.32e4 - 4.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.33e4 - 2.31e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.50e4 + 6.07e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.35e4 - 2.34e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.97e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.17e4 - 3.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.03e5T + 8.58e9T^{2} \)
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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

−13.95172127208168416479548769333, −13.14834119740066171576691001842, −11.89183494484657478897741506402, −10.55273099568533605101876896911, −8.711884023178081049163715470371, −7.78351928278602256665776457931, −6.53172781641714239469902512070, −4.94295543733906248290774680853, −4.14113630618112886663513500779, −0.867141472878998590617865335301, 2.11763828385506265396121672891, 3.31265362931352781397155158593, 4.73458097908058953024985863570, 6.71346355715129733091805075076, 8.040780793052691957782564082760, 9.893283831515083083584705392544, 10.95223612730783218675917488434, 11.90189921264168712488343183786, 12.44624390453632393874494440888, 14.09809795969035264658099080488

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