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Properties

Label 2-369600-1.1-c1-0-141
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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  + 3-s + 7-s + 9-s + 11-s + 3·13-s − 7·19-s + 21-s − 6·23-s + 27-s + 9·29-s + 33-s − 3·37-s + 3·39-s + 8·41-s + 10·43-s − 3·47-s + 49-s + 6·53-s − 7·57-s − 7·59-s − 10·61-s + 63-s − 3·67-s − 6·69-s − 8·71-s + 7·73-s + 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s − 1.60·19-s + 0.218·21-s − 1.25·23-s + 0.192·27-s + 1.67·29-s + 0.174·33-s − 0.493·37-s + 0.480·39-s + 1.24·41-s + 1.52·43-s − 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.927·57-s − 0.911·59-s − 1.28·61-s + 0.125·63-s − 0.366·67-s − 0.722·69-s − 0.949·71-s + 0.819·73-s + 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.415310752\)
\(L(\frac12)\) \(\approx\) \(3.415310752\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.51062002426588, −12.06279595562863, −11.73288409473810, −10.97014178147396, −10.60985044542966, −10.44892350059545, −9.737312902366767, −9.167585056099045, −8.892023163899255, −8.358323892318574, −8.010278933597034, −7.643867713652142, −6.903176943992343, −6.504323817363042, −5.983394251017542, −5.722125738717499, −4.682686935187569, −4.520848121502842, −3.936540172064949, −3.569275638276935, −2.707428983499583, −2.420341063514543, −1.714840321882471, −1.229612702412171, −0.4606386638004739, 0.4606386638004739, 1.229612702412171, 1.714840321882471, 2.420341063514543, 2.707428983499583, 3.569275638276935, 3.936540172064949, 4.520848121502842, 4.682686935187569, 5.722125738717499, 5.983394251017542, 6.504323817363042, 6.903176943992343, 7.643867713652142, 8.010278933597034, 8.358323892318574, 8.892023163899255, 9.167585056099045, 9.737312902366767, 10.44892350059545, 10.60985044542966, 10.97014178147396, 11.73288409473810, 12.06279595562863, 12.51062002426588

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