Properties

Label 2-93600-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 2·11-s − 13-s − 4·17-s − 19-s − 6·23-s − 9·29-s + 4·31-s − 3·37-s − 7·41-s − 4·43-s + 9·47-s − 3·49-s − 53-s + 6·59-s − 6·61-s + 13·67-s + 11·71-s − 8·73-s + 4·77-s − 79-s + 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.970·17-s − 0.229·19-s − 1.25·23-s − 1.67·29-s + 0.718·31-s − 0.493·37-s − 1.09·41-s − 0.609·43-s + 1.31·47-s − 3/7·49-s − 0.137·53-s + 0.781·59-s − 0.768·61-s + 1.58·67-s + 1.30·71-s − 0.936·73-s + 0.455·77-s − 0.112·79-s + 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613486298\)
\(L(\frac12)\) \(\approx\) \(1.613486298\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
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

−13.85060176874867, −13.42844121574093, −12.84008553637796, −12.27768863449600, −11.83992922135035, −11.32916233211716, −11.03293423656489, −10.34437284251684, −9.860187738832283, −9.373572405381681, −8.705535130166808, −8.415304860707914, −7.800184306231435, −7.273724081120627, −6.707736522580355, −6.218743390573136, −5.590390941659843, −5.009772511199413, −4.486535403077561, −3.886921934611114, −3.461250097607803, −2.419332030346222, −1.995699685935769, −1.421265345022610, −0.3916651850037279, 0.3916651850037279, 1.421265345022610, 1.995699685935769, 2.419332030346222, 3.461250097607803, 3.886921934611114, 4.486535403077561, 5.009772511199413, 5.590390941659843, 6.218743390573136, 6.707736522580355, 7.273724081120627, 7.800184306231435, 8.415304860707914, 8.705535130166808, 9.373572405381681, 9.860187738832283, 10.34437284251684, 11.03293423656489, 11.32916233211716, 11.83992922135035, 12.27768863449600, 12.84008553637796, 13.42844121574093, 13.85060176874867

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