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Properties

Label 2-63-21.20-c5-0-10
Degree $2$
Conductor $63$
Sign $-0.504 + 0.863i$
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  − 0.500i·2-s + 31.7·4-s − 63.0·5-s + (−53.6 − 118. i)7-s − 31.9i·8-s + 31.5i·10-s − 215. i·11-s − 204. i·13-s + (−59.1 + 26.8i)14-s + 9.99e2·16-s − 1.94e3·17-s − 2.31e3i·19-s − 2.00e3·20-s − 107.·22-s − 3.83e3i·23-s + ⋯
L(s)  = 1  − 0.0885i·2-s + 0.992·4-s − 1.12·5-s + (−0.414 − 0.910i)7-s − 0.176i·8-s + 0.0999i·10-s − 0.536i·11-s − 0.335i·13-s + (−0.0806 + 0.0366i)14-s + 0.976·16-s − 1.63·17-s − 1.47i·19-s − 1.11·20-s − 0.0475·22-s − 1.51i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.504 + 0.863i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.504 + 0.863i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.571963 - 0.996125i\)
\(L(\frac12)\) \(\approx\) \(0.571963 - 0.996125i\)
\(L(\frac{7}{2})\) 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 \( 1 + (53.6 + 118. i)T \)
good2 \( 1 + 0.500iT - 32T^{2} \)
5 \( 1 + 63.0T + 3.12e3T^{2} \)
11 \( 1 + 215. iT - 1.61e5T^{2} \)
13 \( 1 + 204. iT - 3.71e5T^{2} \)
17 \( 1 + 1.94e3T + 1.41e6T^{2} \)
19 \( 1 + 2.31e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.83e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.57e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.95e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.06e3T + 6.93e7T^{2} \)
41 \( 1 - 7.65e3T + 1.15e8T^{2} \)
43 \( 1 + 5.67e3T + 1.47e8T^{2} \)
47 \( 1 - 1.62e4T + 2.29e8T^{2} \)
53 \( 1 - 2.46e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.64e4T + 7.14e8T^{2} \)
61 \( 1 - 1.97e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.13e4T + 1.35e9T^{2} \)
71 \( 1 + 7.64e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.46e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.89e4T + 3.07e9T^{2} \)
83 \( 1 + 3.79e4T + 3.93e9T^{2} \)
89 \( 1 - 4.24e4T + 5.58e9T^{2} \)
97 \( 1 - 1.00e5iT - 8.58e9T^{2} \)
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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

−13.48105710954567374581124004153, −12.35377563029701409018711289058, −11.13070233247525166661961164673, −10.64168518459154840913748152180, −8.729028407145893952952535034418, −7.36888410854661012092319813919, −6.54605187135470930615101225919, −4.34366987904117239093636521185, −2.88185664015365490019875297192, −0.50348456982078222430521305368, 2.18852073557731460977950317737, 3.89166789721544711246690398887, 5.87532114536495901278604646725, 7.15381339281393756622766802742, 8.244132023451514173654104405052, 9.746752598611282643237677159119, 11.35196905678430311601936119257, 11.83743124529526770284059452218, 12.97217209236613644109777147581, 14.76909127202262436508917179023

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