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Properties

Label 2-92736-1.1-c1-0-76
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  + 7-s − 6·11-s + 3·13-s − 23-s − 5·25-s − 3·29-s − 7·31-s − 8·37-s + 11·41-s − 4·43-s − 47-s + 49-s + 4·53-s + 12·59-s + 6·61-s − 12·67-s + 5·71-s + 15·73-s − 6·77-s − 4·79-s + 12·89-s + 3·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.80·11-s + 0.832·13-s − 0.208·23-s − 25-s − 0.557·29-s − 1.25·31-s − 1.31·37-s + 1.71·41-s − 0.609·43-s − 0.145·47-s + 1/7·49-s + 0.549·53-s + 1.56·59-s + 0.768·61-s − 1.46·67-s + 0.593·71-s + 1.75·73-s − 0.683·77-s − 0.450·79-s + 1.27·89-s + 0.314·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: Trivial
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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.94959557903055, −13.54604270464348, −13.17012204823442, −12.63370891330749, −12.25462261889159, −11.36585767024683, −11.22191456307445, −10.63235819189146, −10.19698378136729, −9.706250971058211, −9.007188522105184, −8.535986045783712, −8.027209754329751, −7.571918706494403, −7.165437389057454, −6.395719214445798, −5.687386088834959, −5.441792569995536, −4.919564264808666, −4.070344813433631, −3.666058467337541, −2.961996659186902, −2.176174915150045, −1.849606743316262, −0.7909625372316762, 0, 0.7909625372316762, 1.849606743316262, 2.176174915150045, 2.961996659186902, 3.666058467337541, 4.070344813433631, 4.919564264808666, 5.441792569995536, 5.687386088834959, 6.395719214445798, 7.165437389057454, 7.571918706494403, 8.027209754329751, 8.535986045783712, 9.007188522105184, 9.706250971058211, 10.19698378136729, 10.63235819189146, 11.22191456307445, 11.36585767024683, 12.25462261889159, 12.63370891330749, 13.17012204823442, 13.54604270464348, 13.94959557903055

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