Properties

Degree $2$
Conductor $66654$
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 − 8-s + 2·10-s − 4·11-s + 6·13-s − 14-s + 16-s + 2·17-s + 4·19-s − 2·20-s + 4·22-s − 25-s − 6·26-s + 28-s + 2·29-s − 32-s − 2·34-s − 2·35-s + 10·37-s − 4·38-s + 2·40-s + 6·41-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 − 0.353·8-s + 0.632·10-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s − 1.17·26-s + 0.188·28-s + 0.371·29-s − 0.176·32-s − 0.342·34-s − 0.338·35-s + 1.64·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66654\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{66654} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 66654,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.673578828\)
\(L(\frac12)\) \(\approx\) \(1.673578828\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
23 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.23406642326220, −13.61125648464874, −13.24564525505032, −12.58956629910517, −12.07401862860343, −11.46543807812430, −11.20831274001832, −10.65829519983033, −10.27075156799812, −9.520279354827383, −9.085484718384670, −8.334948220609833, −8.091883020867864, −7.576193767933165, −7.274016504748076, −6.321682517917712, −5.845670714770711, −5.375891361697304, −4.482613952765056, −4.030064063318675, −3.229114745488114, −2.835250679983705, −1.937248156687447, −1.074500940917027, −0.5824027663472188, 0.5824027663472188, 1.074500940917027, 1.937248156687447, 2.835250679983705, 3.229114745488114, 4.030064063318675, 4.482613952765056, 5.375891361697304, 5.845670714770711, 6.321682517917712, 7.274016504748076, 7.576193767933165, 8.091883020867864, 8.334948220609833, 9.085484718384670, 9.520279354827383, 10.27075156799812, 10.65829519983033, 11.20831274001832, 11.46543807812430, 12.07401862860343, 12.58956629910517, 13.24564525505032, 13.61125648464874, 14.23406642326220

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