Properties

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

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: \(1\)
Selberg data: \((2,\ 92736,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−14.06551032059308, −13.61261879146059, −13.18524673751810, −12.31529067239658, −12.15164238208550, −11.58929885672201, −11.43604137795458, −10.57781075474765, −10.06389567322956, −9.651117851792756, −9.108947964877272, −8.551129819872597, −7.929845325854959, −7.698329479089635, −6.906743484548991, −6.582569020668439, −5.872584954039744, −5.428055740861246, −4.681586423993001, −4.032760444001207, −3.531197255692277, −3.279558041028618, −2.391144198392799, −1.339005790758840, −1.023310109933201, 0, 1.023310109933201, 1.339005790758840, 2.391144198392799, 3.279558041028618, 3.531197255692277, 4.032760444001207, 4.681586423993001, 5.428055740861246, 5.872584954039744, 6.582569020668439, 6.906743484548991, 7.698329479089635, 7.929845325854959, 8.551129819872597, 9.108947964877272, 9.651117851792756, 10.06389567322956, 10.57781075474765, 11.43604137795458, 11.58929885672201, 12.15164238208550, 12.31529067239658, 13.18524673751810, 13.61261879146059, 14.06551032059308

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