Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s − 7-s − 3·8-s + 2·10-s − 4·11-s − 2·13-s − 14-s − 16-s + 6·17-s + 4·19-s − 2·20-s − 4·22-s − 25-s − 2·26-s + 28-s + 2·29-s + 5·32-s + 6·34-s − 2·35-s + 6·37-s + 4·38-s − 6·40-s − 2·41-s − 4·43-s + 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s − 1.06·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.371·29-s + 0.883·32-s + 1.02·34-s − 0.338·35-s + 0.986·37-s + 0.648·38-s − 0.948·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.103310464\)
\(L(\frac12)\) \(\approx\) \(1.103310464\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−18.83789457342104, −18.14196021411947, −17.13317695818883, −15.90301054312043, −14.66963209598428, −13.77909280462587, −13.00543957503168, −12.02428486441375, −10.18331996404874, −9.457705101580371, −7.812954303677477, −5.946076654452876, −5.027094162964051, −3.054220741054584, 3.054220741054584, 5.027094162964051, 5.946076654452876, 7.812954303677477, 9.457705101580371, 10.18331996404874, 12.02428486441375, 13.00543957503168, 13.77909280462587, 14.66963209598428, 15.90301054312043, 17.13317695818883, 18.14196021411947, 18.83789457342104

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