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Properties

Label 2-92400-1.1-c1-0-138
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 11-s − 3·13-s − 7·19-s − 21-s + 6·23-s + 27-s − 9·29-s + 33-s + 3·37-s − 3·39-s + 8·41-s + 10·43-s + 3·47-s + 49-s − 6·53-s − 7·57-s − 7·59-s + 10·61-s − 63-s − 3·67-s + 6·69-s + 8·71-s + 7·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 1.60·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s − 1.67·29-s + 0.174·33-s + 0.493·37-s − 0.480·39-s + 1.24·41-s + 1.52·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.927·57-s − 0.911·59-s + 1.28·61-s − 0.125·63-s − 0.366·67-s + 0.722·69-s + 0.949·71-s + 0.819·73-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: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.31076155654859, −13.57937657426097, −12.90963331422062, −12.68946490577568, −12.47120155183573, −11.46422234388659, −11.14696650379895, −10.65203039538606, −10.04605521649685, −9.498763843901449, −9.054700814849014, −8.810034488992641, −7.978343867830736, −7.503643979317081, −7.150312612312713, −6.411355500907286, −6.054271251932497, −5.272739692754474, −4.722005368494007, −4.006143354209100, −3.759039827631184, −2.753310963905726, −2.488348594669428, −1.766317556354579, −0.8900732572715907, 0, 0.8900732572715907, 1.766317556354579, 2.488348594669428, 2.753310963905726, 3.759039827631184, 4.006143354209100, 4.722005368494007, 5.272739692754474, 6.054271251932497, 6.411355500907286, 7.150312612312713, 7.503643979317081, 7.978343867830736, 8.810034488992641, 9.054700814849014, 9.498763843901449, 10.04605521649685, 10.65203039538606, 11.14696650379895, 11.46422234388659, 12.47120155183573, 12.68946490577568, 12.90963331422062, 13.57937657426097, 14.31076155654859

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