Properties

Label 2-93600-1.1-c1-0-59
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  − 3·7-s − 3·11-s − 13-s − 5·17-s + 6·19-s + 3·23-s + 2·29-s − 6·31-s + 3·37-s − 5·41-s + 2·49-s + 7·53-s + 61-s − 15·71-s − 2·73-s + 9·77-s + 15·79-s + 6·83-s − 11·89-s + 3·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 15·119-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.904·11-s − 0.277·13-s − 1.21·17-s + 1.37·19-s + 0.625·23-s + 0.371·29-s − 1.07·31-s + 0.493·37-s − 0.780·41-s + 2/7·49-s + 0.961·53-s + 0.128·61-s − 1.78·71-s − 0.234·73-s + 1.02·77-s + 1.68·79-s + 0.658·83-s − 1.16·89-s + 0.314·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.37·119-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.87258196506388, −13.50794780220708, −13.08187785798507, −12.76679125901206, −12.13397944136296, −11.62216973387000, −11.13491600548047, −10.47501531097679, −10.21569096698648, −9.475614309799861, −9.248448491608171, −8.664041734360280, −8.010112194028977, −7.413228683170297, −7.007349288232038, −6.525734316927953, −5.875319692208832, −5.319768848333735, −4.874594506297771, −4.151356464162274, −3.470148314175248, −2.948259035022778, −2.486936425262509, −1.684100932486243, −0.7059682146005219, 0, 0.7059682146005219, 1.684100932486243, 2.486936425262509, 2.948259035022778, 3.470148314175248, 4.151356464162274, 4.874594506297771, 5.319768848333735, 5.875319692208832, 6.525734316927953, 7.007349288232038, 7.413228683170297, 8.010112194028977, 8.664041734360280, 9.248448491608171, 9.475614309799861, 10.21569096698648, 10.47501531097679, 11.13491600548047, 11.62216973387000, 12.13397944136296, 12.76679125901206, 13.08187785798507, 13.50794780220708, 13.87258196506388

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