Properties

Label 2-187200-1.1-c1-0-0
Degree $2$
Conductor $187200$
Sign $1$
Analytic cond. $1494.79$
Root an. cond. $38.6626$
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  − 4·11-s − 13-s + 19-s − 4·23-s − 3·29-s − 4·31-s − 5·37-s − 9·41-s − 2·43-s + 3·47-s − 7·49-s − 53-s − 10·59-s − 4·61-s + 9·67-s + 7·71-s − 4·73-s − 11·79-s + 6·83-s − 10·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.277·13-s + 0.229·19-s − 0.834·23-s − 0.557·29-s − 0.718·31-s − 0.821·37-s − 1.40·41-s − 0.304·43-s + 0.437·47-s − 49-s − 0.137·53-s − 1.30·59-s − 0.512·61-s + 1.09·67-s + 0.830·71-s − 0.468·73-s − 1.23·79-s + 0.658·83-s − 1.05·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187200\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1494.79\)
Root analytic conductor: \(38.6626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 187200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08913420048\)
\(L(\frac12)\) \(\approx\) \(0.08913420048\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.02328111336642, −12.74450456277215, −12.20965260896005, −11.74182365414373, −11.22284937977053, −10.74247708958853, −10.28585831479891, −9.873140348167526, −9.424398715605611, −8.763252264594476, −8.360674656175340, −7.760080620964669, −7.503354788588313, −6.866843199856722, −6.357142623251004, −5.694895837477172, −5.310148467927146, −4.833321663697894, −4.272187003426179, −3.442184284721482, −3.234474874258857, −2.358440581071075, −1.945064620222947, −1.245607837973191, −0.08304301364518524, 0.08304301364518524, 1.245607837973191, 1.945064620222947, 2.358440581071075, 3.234474874258857, 3.442184284721482, 4.272187003426179, 4.833321663697894, 5.310148467927146, 5.694895837477172, 6.357142623251004, 6.866843199856722, 7.503354788588313, 7.760080620964669, 8.360674656175340, 8.763252264594476, 9.424398715605611, 9.873140348167526, 10.28585831479891, 10.74247708958853, 11.22284937977053, 11.74182365414373, 12.20965260896005, 12.74450456277215, 13.02328111336642

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