Properties

Degree $2$
Conductor $146523$
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 + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·15-s − 16-s + 18-s + 4·19-s + 2·20-s + 4·22-s − 3·24-s − 25-s + 27-s + 2·29-s − 2·30-s − 8·31-s + 5·32-s + 4·33-s − 36-s − 2·37-s + 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.516·15-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.365·30-s − 1.43·31-s + 0.883·32-s + 0.696·33-s − 1/6·36-s − 0.328·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 146523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 146523 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(146523\)    =    \(3 \cdot 13^{2} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{146523} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 146523,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724606097\)
\(L(\frac12)\) \(\approx\) \(1.724606097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.35865530903204, −13.00649172109934, −12.36406056834694, −12.05721089511280, −11.64910743013071, −11.14691637624011, −10.58214314268574, −9.724187687426785, −9.454606337519330, −9.111012896066730, −8.462162166092363, −8.065145587526494, −7.533371263445458, −7.027304419709981, −6.356900737875643, −5.982968549445337, −5.172155871754265, −4.737947167632555, −4.243553587397301, −3.672887575062311, −3.331512692957827, −2.942888177411667, −1.836168079109716, −1.332221749924573, −0.3405689041122532, 0.3405689041122532, 1.332221749924573, 1.836168079109716, 2.942888177411667, 3.331512692957827, 3.672887575062311, 4.243553587397301, 4.737947167632555, 5.172155871754265, 5.982968549445337, 6.356900737875643, 7.027304419709981, 7.533371263445458, 8.065145587526494, 8.462162166092363, 9.111012896066730, 9.454606337519330, 9.724187687426785, 10.58214314268574, 11.14691637624011, 11.64910743013071, 12.05721089511280, 12.36406056834694, 13.00649172109934, 13.35865530903204

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