Properties

Degree $2$
Conductor $63$
Sign $-0.577 + 0.816i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.40i·2-s − 21.2·4-s − 18.5·7-s − 71.5i·8-s + 66.7i·11-s − 100. i·14-s + 216.·16-s − 361.·22-s + 125. i·23-s − 125·25-s + 393.·28-s + 69.7i·29-s + 600. i·32-s + 10.5·37-s + 534.·43-s − 1.41e3i·44-s + ⋯
L(s)  = 1  + 1.91i·2-s − 2.65·4-s − 0.999·7-s − 3.16i·8-s + 1.83i·11-s − 1.91i·14-s + 3.38·16-s − 3.49·22-s + 1.13i·23-s − 25-s + 2.65·28-s + 0.446i·29-s + 3.31i·32-s + 0.0470·37-s + 1.89·43-s − 4.85i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.577 + 0.816i$
Motivic weight: \(3\)
Character: $\chi_{63} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.333519 - 0.644309i\)
\(L(\frac12)\) \(\approx\) \(0.333519 - 0.644309i\)
\(L(\frac{5}{2})\) 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 \( 1 + 18.5T \)
good2 \( 1 - 5.40iT - 8T^{2} \)
5 \( 1 + 125T^{2} \)
11 \( 1 - 66.7iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 - 125. iT - 1.21e4T^{2} \)
29 \( 1 - 69.7iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 10.5T + 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 534.T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 65.4iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 740T + 3.00e5T^{2} \)
71 \( 1 - 205. iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 + 1.38e3T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−15.44607044552172187309435252730, −14.41324914033663825553491800872, −13.28830523720173095940421607502, −12.41587010489270299267074588621, −9.923477192646726845296226898316, −9.241450780954893282778454900706, −7.65469449645162492878361295795, −6.85882528085845714350427961432, −5.59328794845104549216704373775, −4.14452910051773280477311402234, 0.50827161462858869577201668080, 2.76272598405608119082452877240, 3.93566870972386500702500289583, 5.85680225559852868161634825107, 8.402392327330469460790107586973, 9.420753723152216723632914325286, 10.51009801635924384486448848235, 11.39959027761754554692468351354, 12.49424301303578441900398268378, 13.40611239178433023752164801523

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