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Properties

Label 2-92400-1.1-c1-0-85
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 − 2·13-s − 2·17-s − 4·19-s + 21-s − 4·23-s − 27-s − 6·29-s − 33-s − 2·37-s + 2·39-s + 6·41-s + 12·43-s + 8·47-s + 49-s + 2·51-s + 6·53-s + 4·57-s − 8·59-s − 14·61-s − 63-s + 12·67-s + 4·69-s − 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.04·59-s − 1.79·61-s − 0.125·63-s + 1.46·67-s + 0.481·69-s − 0.949·71-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.95214114005144, −13.67954428228578, −12.87697359218852, −12.65728836421617, −12.13354544058306, −11.69919371169247, −11.07839973826171, −10.55418538858637, −10.37261409918712, −9.485204610676700, −9.199512468773179, −8.763737929431599, −7.856028443604193, −7.533730819158685, −6.979231899482974, −6.326415493224465, −5.956211638423863, −5.484391363415865, −4.700923455440997, −4.154766011925820, −3.845775226188694, −2.875315477787677, −2.280347483848240, −1.681725167703145, −0.7075798230521380, 0, 0.7075798230521380, 1.681725167703145, 2.280347483848240, 2.875315477787677, 3.845775226188694, 4.154766011925820, 4.700923455440997, 5.484391363415865, 5.956211638423863, 6.326415493224465, 6.979231899482974, 7.533730819158685, 7.856028443604193, 8.763737929431599, 9.199512468773179, 9.485204610676700, 10.37261409918712, 10.55418538858637, 11.07839973826171, 11.69919371169247, 12.13354544058306, 12.65728836421617, 12.87697359218852, 13.67954428228578, 13.95214114005144

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