Properties

Label 2-187200-1.1-c1-0-15
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  − 2·7-s + 2·11-s − 13-s − 2·17-s − 2·19-s − 8·23-s − 6·29-s − 2·31-s + 2·37-s + 2·41-s + 6·47-s − 3·49-s + 10·53-s − 14·59-s + 10·61-s + 2·67-s + 6·71-s − 2·73-s − 4·77-s − 12·79-s − 6·83-s + 18·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s − 1.66·23-s − 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.312·41-s + 0.875·47-s − 3/7·49-s + 1.37·53-s − 1.82·59-s + 1.28·61-s + 0.244·67-s + 0.712·71-s − 0.234·73-s − 0.455·77-s − 1.35·79-s − 0.658·83-s + 1.90·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(0.7150371390\)
\(L(\frac12)\) \(\approx\) \(0.7150371390\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.08995151600556, −12.70316827924520, −12.17036774609701, −11.77287485027514, −11.31070075247012, −10.71485576093195, −10.27481310834470, −9.798820985237851, −9.303569218327328, −8.978417660458138, −8.405415052835719, −7.735149989493755, −7.457207034603121, −6.601112445195715, −6.532311793248655, −5.762372583612470, −5.503575288060522, −4.616996147469716, −4.085651675377716, −3.776717660524260, −3.117660672134683, −2.330914186303811, −2.005512526458107, −1.163002162075334, −0.2464236385882396, 0.2464236385882396, 1.163002162075334, 2.005512526458107, 2.330914186303811, 3.117660672134683, 3.776717660524260, 4.085651675377716, 4.616996147469716, 5.503575288060522, 5.762372583612470, 6.532311793248655, 6.601112445195715, 7.457207034603121, 7.735149989493755, 8.405415052835719, 8.978417660458138, 9.303569218327328, 9.798820985237851, 10.27481310834470, 10.71485576093195, 11.31070075247012, 11.77287485027514, 12.17036774609701, 12.70316827924520, 13.08995151600556

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