Properties

Label 2-93600-1.1-c1-0-10
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 + 4·11-s − 13-s − 6·17-s − 6·19-s + 2·29-s + 6·31-s − 10·37-s − 8·41-s + 12·43-s − 12·47-s − 3·49-s − 6·53-s + 2·61-s + 2·67-s − 8·71-s − 14·73-s + 8·77-s − 4·79-s − 8·83-s − 4·89-s − 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 1.37·19-s + 0.371·29-s + 1.07·31-s − 1.64·37-s − 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.256·61-s + 0.244·67-s − 0.949·71-s − 1.63·73-s + 0.911·77-s − 0.450·79-s − 0.878·83-s − 0.423·89-s − 0.209·91-s − 1.42·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.638202837\)
\(L(\frac12)\) \(\approx\) \(1.638202837\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.97709599710834, −13.28702346031872, −12.90783966853489, −12.25972991825016, −11.85790878880403, −11.31248655578064, −11.02420406706518, −10.34040872049776, −9.966385157712933, −9.196185004011412, −8.756963237717244, −8.449521624538284, −7.913429351844918, −7.072740641705274, −6.712733504306135, −6.325724711832618, −5.659552617585815, −4.825770940495532, −4.484704803693864, −4.102624895539043, −3.271458419425881, −2.598290396469973, −1.760182302849491, −1.556490031359764, −0.3896477726043913, 0.3896477726043913, 1.556490031359764, 1.760182302849491, 2.598290396469973, 3.271458419425881, 4.102624895539043, 4.484704803693864, 4.825770940495532, 5.659552617585815, 6.325724711832618, 6.712733504306135, 7.072740641705274, 7.913429351844918, 8.449521624538284, 8.756963237717244, 9.196185004011412, 9.966385157712933, 10.34040872049776, 11.02420406706518, 11.31248655578064, 11.85790878880403, 12.25972991825016, 12.90783966853489, 13.28702346031872, 13.97709599710834

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