Properties

Label 2-187200-1.1-c1-0-286
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 4·11-s − 13-s + 8·17-s − 6·19-s − 6·23-s − 4·29-s − 2·37-s + 2·41-s + 4·43-s − 3·49-s + 10·53-s − 4·59-s + 10·61-s − 12·67-s − 8·71-s + 8·73-s − 8·77-s − 8·79-s + 12·83-s + 14·89-s − 2·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.20·11-s − 0.277·13-s + 1.94·17-s − 1.37·19-s − 1.25·23-s − 0.742·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s + 1.37·53-s − 0.520·59-s + 1.28·61-s − 1.46·67-s − 0.949·71-s + 0.936·73-s − 0.911·77-s − 0.900·79-s + 1.31·83-s + 1.48·89-s − 0.209·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

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

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: $\chi_{187200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 187200,\ (\ :1/2),\ -1)\)

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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.36532249192694, −12.78367215171711, −12.49696952407427, −11.87722235141943, −11.64410390367267, −10.86729607670582, −10.47347592258729, −10.22019299367690, −9.677312771738294, −9.059074635035213, −8.444551287560496, −8.079783512126297, −7.563821699263142, −7.435819750005909, −6.521461009316223, −5.943983093258525, −5.532542878031436, −5.110151511716627, −4.502206901788876, −3.945680717199900, −3.399622502409603, −2.671780838908629, −2.139790484097724, −1.633114571899195, −0.7699884000748574, 0, 0.7699884000748574, 1.633114571899195, 2.139790484097724, 2.671780838908629, 3.399622502409603, 3.945680717199900, 4.502206901788876, 5.110151511716627, 5.532542878031436, 5.943983093258525, 6.521461009316223, 7.435819750005909, 7.563821699263142, 8.079783512126297, 8.444551287560496, 9.059074635035213, 9.677312771738294, 10.22019299367690, 10.47347592258729, 10.86729607670582, 11.64410390367267, 11.87722235141943, 12.49696952407427, 12.78367215171711, 13.36532249192694

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