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Properties

Label 2-92400-1.1-c1-0-74
Degree $2$
Conductor $92400$
Sign $1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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  + 3-s + 7-s + 9-s − 11-s + 6·13-s + 2·17-s + 2·19-s + 21-s + 27-s − 2·29-s + 4·31-s − 33-s − 8·37-s + 6·39-s + 6·41-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 2·57-s + 10·59-s − 4·61-s + 63-s − 2·67-s − 4·71-s − 10·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s − 1.31·37-s + 0.960·39-s + 0.937·41-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.264·57-s + 1.30·59-s − 0.512·61-s + 0.125·63-s − 0.244·67-s − 0.474·71-s − 1.17·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.157332676\)
\(L(\frac12)\) \(\approx\) \(4.157332676\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−13.96369150019694, −13.36156972393551, −13.01783774103202, −12.39238431380360, −11.88695829868048, −11.30810104549631, −10.91413985810659, −10.31154527615413, −9.972187472398526, −9.170740298558270, −8.838299928825996, −8.344492349564118, −7.882832174038003, −7.343334918959558, −6.824046931711160, −6.090069380545711, −5.654769459363909, −5.126879160996270, −4.241112383390548, −3.981293023498853, −3.163598290143494, −2.837595057711713, −1.868033176915555, −1.378504856005403, −0.6535086198674231, 0.6535086198674231, 1.378504856005403, 1.868033176915555, 2.837595057711713, 3.163598290143494, 3.981293023498853, 4.241112383390548, 5.126879160996270, 5.654769459363909, 6.090069380545711, 6.824046931711160, 7.343334918959558, 7.882832174038003, 8.344492349564118, 8.838299928825996, 9.170740298558270, 9.972187472398526, 10.31154527615413, 10.91413985810659, 11.30810104549631, 11.88695829868048, 12.39238431380360, 13.01783774103202, 13.36156972393551, 13.96369150019694

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