Properties

Label 2-93600-1.1-c1-0-49
Degree $2$
Conductor $93600$
Sign $-1$
Analytic cond. $747.399$
Root an. cond. $27.3386$
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  − 4·7-s − 4·11-s + 13-s + 2·17-s − 4·19-s − 6·29-s − 10·37-s − 2·41-s + 4·43-s + 4·47-s + 9·49-s − 6·53-s + 4·59-s − 2·61-s + 4·67-s + 8·71-s + 2·73-s + 16·77-s + 16·79-s + 4·83-s − 10·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 1.11·29-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 0.234·73-s + 1.82·77-s + 1.80·79-s + 0.439·83-s − 1.05·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93600\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(747.399\)
Root analytic conductor: \(27.3386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{93600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.97067688975865, −13.45848564765773, −13.06602029724138, −12.61482424650820, −12.34897753353009, −11.68290306459442, −10.80639063175437, −10.73106867456340, −10.11354178672608, −9.623882000161584, −9.179869053072734, −8.597086979949117, −8.043431947419155, −7.501769381993879, −6.936735685019603, −6.444755805004577, −5.933351870796218, −5.363187350729804, −4.919730423745787, −3.937516012248221, −3.617412750382246, −2.994633107882670, −2.395765505306871, −1.759118778793033, −0.6348922765239105, 0, 0.6348922765239105, 1.759118778793033, 2.395765505306871, 2.994633107882670, 3.617412750382246, 3.937516012248221, 4.919730423745787, 5.363187350729804, 5.933351870796218, 6.444755805004577, 6.936735685019603, 7.501769381993879, 8.043431947419155, 8.597086979949117, 9.179869053072734, 9.623882000161584, 10.11354178672608, 10.73106867456340, 10.80639063175437, 11.68290306459442, 12.34897753353009, 12.61482424650820, 13.06602029724138, 13.45848564765773, 13.97067688975865

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