Properties

Label 2-92736-1.1-c1-0-135
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  + 3·5-s + 7-s − 4·11-s + 3·13-s + 4·17-s + 23-s + 4·25-s + 3·29-s + 6·31-s + 3·35-s + 9·37-s − 9·41-s − 3·43-s − 7·47-s + 49-s − 4·53-s − 12·55-s − 6·59-s − 10·61-s + 9·65-s + 4·67-s − 6·71-s − 8·73-s − 4·77-s − 8·79-s − 4·83-s + 12·85-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 1.20·11-s + 0.832·13-s + 0.970·17-s + 0.208·23-s + 4/5·25-s + 0.557·29-s + 1.07·31-s + 0.507·35-s + 1.47·37-s − 1.40·41-s − 0.457·43-s − 1.02·47-s + 1/7·49-s − 0.549·53-s − 1.61·55-s − 0.781·59-s − 1.28·61-s + 1.11·65-s + 0.488·67-s − 0.712·71-s − 0.936·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-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: $-1$
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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.92018467857116, −13.47919791210261, −13.31360477620658, −12.78589752436097, −12.05383244789968, −11.72483198915308, −10.89909681941277, −10.63363497237948, −10.08096788375674, −9.705064191150680, −9.246967288337048, −8.381096240384981, −8.211291340403477, −7.641655306719905, −6.892747678135333, −6.296218138044984, −5.899137064515182, −5.407765052935314, −4.856599148543706, −4.393202744927612, −3.366683449564583, −2.906013869760671, −2.365983083450185, −1.486674347893377, −1.203094841913124, 0, 1.203094841913124, 1.486674347893377, 2.365983083450185, 2.906013869760671, 3.366683449564583, 4.393202744927612, 4.856599148543706, 5.407765052935314, 5.899137064515182, 6.296218138044984, 6.892747678135333, 7.641655306719905, 8.211291340403477, 8.381096240384981, 9.246967288337048, 9.705064191150680, 10.08096788375674, 10.63363497237948, 10.89909681941277, 11.72483198915308, 12.05383244789968, 12.78589752436097, 13.31360477620658, 13.47919791210261, 13.92018467857116

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