Properties

Degree $2$
Conductor $63$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s + 18·5-s + 7·7-s − 21·8-s + 54·10-s + 36·11-s − 34·13-s + 21·14-s − 71·16-s − 42·17-s − 124·19-s + 18·20-s + 108·22-s + 199·25-s − 102·26-s + 7·28-s − 102·29-s − 160·31-s − 45·32-s − 126·34-s + 126·35-s + 398·37-s − 372·38-s − 378·40-s + 318·41-s − 268·43-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s + 1.60·5-s + 0.377·7-s − 0.928·8-s + 1.70·10-s + 0.986·11-s − 0.725·13-s + 0.400·14-s − 1.10·16-s − 0.599·17-s − 1.49·19-s + 0.201·20-s + 1.04·22-s + 1.59·25-s − 0.769·26-s + 0.0472·28-s − 0.653·29-s − 0.926·31-s − 0.248·32-s − 0.635·34-s + 0.608·35-s + 1.76·37-s − 1.58·38-s − 1.49·40-s + 1.21·41-s − 0.950·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{63} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.58059\)
\(L(\frac12)\) \(\approx\) \(2.58059\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
good2 \( 1 - 3 T + p^{3} T^{2} \)
5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 + 34 T + p^{3} T^{2} \)
17 \( 1 + 42 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + 102 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 - 398 T + p^{3} T^{2} \)
41 \( 1 - 318 T + p^{3} T^{2} \)
43 \( 1 + 268 T + p^{3} T^{2} \)
47 \( 1 + 240 T + p^{3} T^{2} \)
53 \( 1 - 498 T + p^{3} T^{2} \)
59 \( 1 - 132 T + p^{3} T^{2} \)
61 \( 1 - 398 T + p^{3} T^{2} \)
67 \( 1 - 92 T + p^{3} T^{2} \)
71 \( 1 - 720 T + p^{3} T^{2} \)
73 \( 1 + 502 T + p^{3} T^{2} \)
79 \( 1 + 1024 T + p^{3} T^{2} \)
83 \( 1 - 204 T + p^{3} T^{2} \)
89 \( 1 + 354 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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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

−14.49829183826360221527171609682, −13.34846620815021391407273648278, −12.70365856960908057710330784969, −11.28247764789895742476523891936, −9.770577806245046420715575347906, −8.861546314641648870266197555738, −6.61089572895162571401593617680, −5.62491856567127410721793378100, −4.32018468037435003529221152115, −2.23150068857170534649981502500, 2.23150068857170534649981502500, 4.32018468037435003529221152115, 5.62491856567127410721793378100, 6.61089572895162571401593617680, 8.861546314641648870266197555738, 9.770577806245046420715575347906, 11.28247764789895742476523891936, 12.70365856960908057710330784969, 13.34846620815021391407273648278, 14.49829183826360221527171609682

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