Properties

Degree $2$
Conductor $92736$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s − 4·19-s + 23-s − 25-s − 2·29-s − 8·31-s − 2·35-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s + 8·55-s + 4·59-s + 10·61-s − 4·65-s − 4·67-s + 8·71-s − 6·73-s − 4·77-s − 12·83-s − 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 0.702·73-s − 0.455·77-s − 1.31·83-s − 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92736\)    =    \(2^{6} \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{92736} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.351560335\)
\(L(\frac12)\) \(\approx\) \(1.351560335\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.88835982643853, −13.17269082915420, −12.88671866512394, −12.35916997285268, −11.88790861532318, −11.36140684242406, −10.79344006714452, −10.56818417713680, −9.974834205889178, −9.309262519863630, −8.648522175060916, −8.339557706263207, −7.704065353831256, −7.420504834904778, −6.955284100520333, −5.955043265271207, −5.612387675690604, −5.158084474170793, −4.384508479602707, −3.809281206208688, −3.476483775039459, −2.615947782103817, −2.063308555442323, −1.179772394524797, −0.3973652993486306, 0.3973652993486306, 1.179772394524797, 2.063308555442323, 2.615947782103817, 3.476483775039459, 3.809281206208688, 4.384508479602707, 5.158084474170793, 5.612387675690604, 5.955043265271207, 6.955284100520333, 7.420504834904778, 7.704065353831256, 8.339557706263207, 8.648522175060916, 9.309262519863630, 9.974834205889178, 10.56818417713680, 10.79344006714452, 11.36140684242406, 11.88790861532318, 12.35916997285268, 12.88671866512394, 13.17269082915420, 13.88835982643853

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