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Properties

Label 2-63-7.5-c8-0-14
Degree $2$
Conductor $63$
Sign $0.845 + 0.534i$
Analytic cond. $25.6648$
Root an. cond. $5.06604$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.6 − 21.9i)2-s + (−193. − 334. i)4-s + (538. + 310. i)5-s + (207. + 2.39e3i)7-s − 3.29e3·8-s + (1.36e4 − 7.87e3i)10-s + (7.54e3 + 1.30e4i)11-s + 4.51e4i·13-s + (5.51e4 + 2.57e4i)14-s + (7.65e3 − 1.32e4i)16-s + (−3.59e4 + 2.07e4i)17-s + (1.02e5 + 5.90e4i)19-s − 2.40e5i·20-s + 3.82e5·22-s + (1.08e5 − 1.87e5i)23-s + ⋯
L(s)  = 1  + (0.791 − 1.37i)2-s + (−0.754 − 1.30i)4-s + (0.861 + 0.497i)5-s + (0.0864 + 0.996i)7-s − 0.804·8-s + (1.36 − 0.787i)10-s + (0.515 + 0.892i)11-s + 1.57i·13-s + (1.43 + 0.670i)14-s + (0.116 − 0.202i)16-s + (−0.430 + 0.248i)17-s + (0.784 + 0.453i)19-s − 1.50i·20-s + 1.63·22-s + (0.387 − 0.671i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.845 + 0.534i$
Analytic conductor: \(25.6648\)
Root analytic conductor: \(5.06604\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :4),\ 0.845 + 0.534i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.34201 - 0.967925i\)
\(L(\frac12)\) \(\approx\) \(3.34201 - 0.967925i\)
\(L(5)\) 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 \( 1 + (-207. - 2.39e3i)T \)
good2 \( 1 + (-12.6 + 21.9i)T + (-128 - 221. i)T^{2} \)
5 \( 1 + (-538. - 310. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-7.54e3 - 1.30e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 4.51e4iT - 8.15e8T^{2} \)
17 \( 1 + (3.59e4 - 2.07e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.02e5 - 5.90e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.08e5 + 1.87e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 3.02e4T + 5.00e11T^{2} \)
31 \( 1 + (1.11e6 - 6.42e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.31e6 + 2.27e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 1.05e6iT - 7.98e12T^{2} \)
43 \( 1 + 6.68e5T + 1.16e13T^{2} \)
47 \( 1 + (8.46e5 + 4.88e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (2.36e5 + 4.08e5i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.62e7 + 9.35e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-2.08e7 - 1.20e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-5.58e6 - 9.67e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 7.52e6T + 6.45e14T^{2} \)
73 \( 1 + (-8.54e6 + 4.93e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (7.33e6 - 1.27e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 3.82e6iT - 2.25e15T^{2} \)
89 \( 1 + (3.43e7 + 1.98e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 7.73e7iT - 7.83e15T^{2} \)
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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

−12.93623991812193229763980631981, −12.04582620460197633134546057164, −11.18579012594517053117231797642, −9.918190182064657912807265299187, −9.103776120652204326682215530359, −6.78246599722780345967850557015, −5.38467798891946454687488226129, −4.03530483150811415465039762785, −2.39827209571471353275837997701, −1.72225293813018319667410282821, 0.954476813159799845151682239632, 3.50393589531968725379404224560, 5.03724816952544612634179635793, 5.88942732584488903619599836879, 7.16215565848197060992523608279, 8.266057399292034484885656701938, 9.704286600892557905540071718197, 11.16390901554397447634386882541, 13.14259550630960526497338645880, 13.37103973282236667070278722139

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