Properties

Label 2-92736-1.1-c1-0-98
Degree $2$
Conductor $92736$
Sign $-1$
Analytic cond. $740.500$
Root an. cond. $27.2121$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 2·11-s + 13-s + 6·17-s + 6·19-s − 23-s − 4·25-s − 5·29-s − 8·31-s − 35-s − 3·37-s + 9·41-s − 3·43-s − 9·47-s + 49-s + 6·53-s − 2·55-s + 8·59-s − 2·61-s + 65-s − 4·67-s + 10·71-s + 2·73-s + 2·77-s − 4·79-s − 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.603·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s − 0.208·23-s − 4/5·25-s − 0.928·29-s − 1.43·31-s − 0.169·35-s − 0.493·37-s + 1.40·41-s − 0.457·43-s − 1.31·47-s + 1/7·49-s + 0.824·53-s − 0.269·55-s + 1.04·59-s − 0.256·61-s + 0.124·65-s − 0.488·67-s + 1.18·71-s + 0.234·73-s + 0.227·77-s − 0.450·79-s − 0.439·83-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: $-1$
Analytic conductor: \(740.500\)
Root analytic conductor: \(27.2121\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{92736} (1, \cdot )$
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 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.18197854667652, −13.52036153209509, −13.12995599595493, −12.68173703314970, −12.16447568605393, −11.59073619250528, −11.18556346896324, −10.56154146225121, −9.958084493659100, −9.706630247181483, −9.275485998571078, −8.576882860669770, −7.983156167145881, −7.402318988846934, −7.254965695552319, −6.302683005772223, −5.796490369409648, −5.401682103841488, −5.039842820524471, −3.989553790046823, −3.550515257163759, −3.071794321428808, −2.287699262206306, −1.642082658808555, −0.9306013534199568, 0, 0.9306013534199568, 1.642082658808555, 2.287699262206306, 3.071794321428808, 3.550515257163759, 3.989553790046823, 5.039842820524471, 5.401682103841488, 5.796490369409648, 6.302683005772223, 7.254965695552319, 7.402318988846934, 7.983156167145881, 8.576882860669770, 9.275485998571078, 9.706630247181483, 9.958084493659100, 10.56154146225121, 11.18556346896324, 11.59073619250528, 12.16447568605393, 12.68173703314970, 13.12995599595493, 13.52036153209509, 14.18197854667652

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