Properties

Label 2-92736-1.1-c1-0-84
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 7-s + 3·11-s + 2·13-s − 4·17-s + 7·19-s + 23-s + 11·25-s + 10·29-s − 2·31-s + 4·35-s + 10·37-s − 7·41-s + 4·43-s − 3·47-s + 49-s − 9·53-s + 12·55-s + 9·59-s + 61-s + 8·65-s − 8·71-s + 4·73-s + 3·77-s + 14·79-s − 6·83-s − 16·85-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 0.970·17-s + 1.60·19-s + 0.208·23-s + 11/5·25-s + 1.85·29-s − 0.359·31-s + 0.676·35-s + 1.64·37-s − 1.09·41-s + 0.609·43-s − 0.437·47-s + 1/7·49-s − 1.23·53-s + 1.61·55-s + 1.17·59-s + 0.128·61-s + 0.992·65-s − 0.949·71-s + 0.468·73-s + 0.341·77-s + 1.57·79-s − 0.658·83-s − 1.73·85-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: \(0\)
Selberg data: \((2,\ 92736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.885576548\)
\(L(\frac12)\) \(\approx\) \(5.885576548\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.86310071737496, −13.37927607364226, −13.09048010132426, −12.40828484655767, −11.78550392638485, −11.41153530845025, −10.79393144528432, −10.36306282476186, −9.741380812447371, −9.343778609443861, −9.093856848000781, −8.352441489405846, −7.947527217040956, −6.949729762428437, −6.686437995222855, −6.215436034091431, −5.627464241475707, −5.153299708273924, −4.595982597724187, −3.966188101300949, −3.047790442313873, −2.666370925861931, −1.869310454448213, −1.322033961463087, −0.8435342658048839, 0.8435342658048839, 1.322033961463087, 1.869310454448213, 2.666370925861931, 3.047790442313873, 3.966188101300949, 4.595982597724187, 5.153299708273924, 5.627464241475707, 6.215436034091431, 6.686437995222855, 6.949729762428437, 7.947527217040956, 8.352441489405846, 9.093856848000781, 9.343778609443861, 9.741380812447371, 10.36306282476186, 10.79393144528432, 11.41153530845025, 11.78550392638485, 12.40828484655767, 13.09048010132426, 13.37927607364226, 13.86310071737496

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