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Properties

Label 2-93600-1.1-c1-0-104
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  + 7-s + 5·11-s + 13-s − 3·17-s − 4·23-s − 9·29-s + 7·31-s − 8·37-s + 2·41-s + 7·47-s − 6·49-s + 3·53-s − 9·59-s − 15·61-s + 7·67-s − 4·73-s + 5·77-s + 8·79-s + 7·83-s + 12·89-s + 91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.50·11-s + 0.277·13-s − 0.727·17-s − 0.834·23-s − 1.67·29-s + 1.25·31-s − 1.31·37-s + 0.312·41-s + 1.02·47-s − 6/7·49-s + 0.412·53-s − 1.17·59-s − 1.92·61-s + 0.855·67-s − 0.468·73-s + 0.569·77-s + 0.900·79-s + 0.768·83-s + 1.27·89-s + 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: \(1\)
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 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.06548690201023, −13.68579194284557, −13.17474690562133, −12.49181305807337, −11.99518482772127, −11.73784136108950, −11.09483261982989, −10.73681010390435, −10.12413287336348, −9.495372094495044, −8.936970116346591, −8.869546871004749, −7.854942620671008, −7.737113172741506, −6.805371858710122, −6.521550092072380, −5.966432999332297, −5.382421285324591, −4.624421684676348, −4.194233694750345, −3.669237888180557, −3.074362190356057, −2.087320930654427, −1.722987394285413, −0.9672103210547174, 0, 0.9672103210547174, 1.722987394285413, 2.087320930654427, 3.074362190356057, 3.669237888180557, 4.194233694750345, 4.624421684676348, 5.382421285324591, 5.966432999332297, 6.521550092072380, 6.805371858710122, 7.737113172741506, 7.854942620671008, 8.869546871004749, 8.936970116346591, 9.495372094495044, 10.12413287336348, 10.73681010390435, 11.09483261982989, 11.73784136108950, 11.99518482772127, 12.49181305807337, 13.17474690562133, 13.68579194284557, 14.06548690201023

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