Properties

Degree $2$
Conductor $63$
Sign $0.313 - 0.949i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + 21.0·5-s + (−11 − 14.8i)7-s + 22.6i·8-s + 59.5i·10-s + 15.5i·11-s + 29.7i·13-s + (42.1 − 31.1i)14-s − 64.0·16-s − 63.2·17-s − 89.3i·19-s − 44·22-s − 77.7i·23-s + 318.·25-s − 84.2·26-s + ⋯
L(s)  = 1  + 0.999i·2-s + 1.88·5-s + (−0.593 − 0.804i)7-s + 0.999i·8-s + 1.88i·10-s + 0.426i·11-s + 0.635i·13-s + (0.804 − 0.593i)14-s − 1.00·16-s − 0.901·17-s − 1.07i·19-s − 0.426·22-s − 0.705i·23-s + 2.55·25-s − 0.635·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.313 - 0.949i$
Motivic weight: \(3\)
Character: $\chi_{63} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.313 - 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.51913 + 1.09768i\)
\(L(\frac12)\) \(\approx\) \(1.51913 + 1.09768i\)
\(L(\frac{5}{2})\) 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 \( 1 + (11 + 14.8i)T \)
good2 \( 1 - 2.82iT - 8T^{2} \)
5 \( 1 - 21.0T + 125T^{2} \)
11 \( 1 - 15.5iT - 1.33e3T^{2} \)
13 \( 1 - 29.7iT - 2.19e3T^{2} \)
17 \( 1 + 63.2T + 4.91e3T^{2} \)
19 \( 1 + 89.3iT - 6.85e3T^{2} \)
23 \( 1 + 77.7iT - 1.21e4T^{2} \)
29 \( 1 - 125. iT - 2.43e4T^{2} \)
31 \( 1 + 238. iT - 2.97e4T^{2} \)
37 \( 1 + 184T + 5.06e4T^{2} \)
41 \( 1 + 105.T + 6.89e4T^{2} \)
43 \( 1 + 190T + 7.95e4T^{2} \)
47 \( 1 - 42.1T + 1.03e5T^{2} \)
53 \( 1 + 357. iT - 1.48e5T^{2} \)
59 \( 1 - 84.2T + 2.05e5T^{2} \)
61 \( 1 + 655. iT - 2.26e5T^{2} \)
67 \( 1 - 296T + 3.00e5T^{2} \)
71 \( 1 - 329. iT - 3.57e5T^{2} \)
73 \( 1 - 804. iT - 3.89e5T^{2} \)
79 \( 1 - 836T + 4.93e5T^{2} \)
83 \( 1 + 1.22e3T + 5.71e5T^{2} \)
89 \( 1 - 695.T + 7.04e5T^{2} \)
97 \( 1 + 566. iT - 9.12e5T^{2} \)
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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

−14.60101847082042716145166891302, −13.75320008599549206999882335838, −12.98935046920190253097894162576, −11.05658252555360789391989479086, −9.904028001666633793903900639830, −8.865301933401142896318093669180, −6.93601546209825179214630967095, −6.41624307458562817545313799155, −4.98468889737987714349882020274, −2.21729374725793309533415613459, 1.78433107398828365457187679253, 3.03008392180011171523769046824, 5.58651069511856230021107528563, 6.54871305090730183721877281609, 8.903497997729015406827512508557, 9.886499631655229577787961530760, 10.61877278634734939034060198100, 12.08992879289533825447468744239, 13.03995073869062088336561097867, 13.80458747573348880544563576091

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