Properties

Degree $2$
Conductor $97461$
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 − 3·8-s − 2·10-s − 4·11-s − 13-s − 16-s + 17-s + 4·19-s + 2·20-s − 4·22-s − 25-s − 26-s + 2·29-s + 8·31-s + 5·32-s + 34-s − 2·37-s + 4·38-s + 6·40-s + 2·41-s − 4·43-s + 4·44-s + 8·47-s − 50-s + 52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.196·26-s + 0.371·29-s + 1.43·31-s + 0.883·32-s + 0.171·34-s − 0.328·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.603·44-s + 1.16·47-s − 0.141·50-s + 0.138·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{97461} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.398673721\)
\(L(\frac12)\) \(\approx\) \(1.398673721\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
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

−13.69160784998694, −13.44424007641807, −12.77841721777142, −12.34586448287869, −11.87486987451510, −11.64003537186510, −10.88959609590332, −10.24185195915717, −9.974950204151524, −9.301210642182273, −8.703226144957044, −8.239963231001737, −7.741231319492237, −7.353916158708218, −6.658694576974186, −5.907498267470411, −5.514117526269426, −4.920811555357698, −4.514205223085029, −3.945505665013343, −3.310997302640097, −2.871052326682088, −2.257190680806315, −1.055091077182486, −0.3863036133917956, 0.3863036133917956, 1.055091077182486, 2.257190680806315, 2.871052326682088, 3.310997302640097, 3.945505665013343, 4.514205223085029, 4.920811555357698, 5.514117526269426, 5.907498267470411, 6.658694576974186, 7.353916158708218, 7.741231319492237, 8.239963231001737, 8.703226144957044, 9.301210642182273, 9.974950204151524, 10.24185195915717, 10.88959609590332, 11.64003537186510, 11.87486987451510, 12.34586448287869, 12.77841721777142, 13.44424007641807, 13.69160784998694

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