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Properties

Label 2-63-21.20-c5-0-1
Degree $2$
Conductor $63$
Sign $0.685 - 0.727i$
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  − 7.98i·2-s − 31.7·4-s − 67.9·5-s + (25.6 + 127. i)7-s − 2.00i·8-s + 542. i·10-s + 181. i·11-s + 796. i·13-s + (1.01e3 − 205. i)14-s − 1.03e3·16-s + 59.3·17-s − 966. i·19-s + 2.15e3·20-s + 1.44e3·22-s + 2.49e3i·23-s + ⋯
L(s)  = 1  − 1.41i·2-s − 0.992·4-s − 1.21·5-s + (0.198 + 0.980i)7-s − 0.0110i·8-s + 1.71i·10-s + 0.451i·11-s + 1.30i·13-s + (1.38 − 0.279i)14-s − 1.00·16-s + 0.0498·17-s − 0.614i·19-s + 1.20·20-s + 0.637·22-s + 0.982i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.685 - 0.727i$
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: \((2,\ 63,\ (\ :5/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.530009 + 0.228763i\)
\(L(\frac12)\) \(\approx\) \(0.530009 + 0.228763i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-25.6 - 127. i)T \)
good2 \( 1 + 7.98iT - 32T^{2} \)
5 \( 1 + 67.9T + 3.12e3T^{2} \)
11 \( 1 - 181. iT - 1.61e5T^{2} \)
13 \( 1 - 796. iT - 3.71e5T^{2} \)
17 \( 1 - 59.3T + 1.41e6T^{2} \)
19 \( 1 + 966. iT - 2.47e6T^{2} \)
23 \( 1 - 2.49e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.78e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.42e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.26e4T + 6.93e7T^{2} \)
41 \( 1 + 1.97e4T + 1.15e8T^{2} \)
43 \( 1 - 1.37e4T + 1.47e8T^{2} \)
47 \( 1 + 2.34e4T + 2.29e8T^{2} \)
53 \( 1 - 2.70e4iT - 4.18e8T^{2} \)
59 \( 1 - 9.08e3T + 7.14e8T^{2} \)
61 \( 1 + 2.24e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.95e4T + 1.35e9T^{2} \)
71 \( 1 - 1.16e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.89e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.52e4T + 3.07e9T^{2} \)
83 \( 1 + 6.45e4T + 3.93e9T^{2} \)
89 \( 1 - 1.03e5T + 5.58e9T^{2} \)
97 \( 1 + 1.45e4iT - 8.58e9T^{2} \)
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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

−13.83630307141518913484075312657, −12.40135027957109533398903447188, −11.82168824597398339757067279635, −11.13242077590995540790483454468, −9.642383612397138352954273808554, −8.582827040633406778662781882861, −6.97093841182514600470086762802, −4.72336136707475224002748572135, −3.41754901480310806891735600514, −1.83298449805950142665771517312, 0.26681910474026482952579214774, 3.71077479789247721467898710194, 5.17547361120411653801869902733, 6.72910791738134862518281214843, 7.81325618248116183897861143689, 8.377584673155523238537109647775, 10.38928583713844769017609836770, 11.53532908136244208342697757823, 12.99969875222329254935175797295, 14.24279412026790583260813934480

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