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Properties

Label 2-135240-1.1-c1-0-4
Degree $2$
Conductor $135240$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
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 − 5-s + 9-s − 2·11-s + 2·13-s + 15-s − 8·17-s + 2·19-s + 23-s + 25-s − 27-s − 8·29-s + 10·31-s + 2·33-s − 2·37-s − 2·39-s − 2·41-s − 2·43-s − 45-s − 6·47-s + 8·51-s + 2·53-s + 2·55-s − 2·57-s − 12·59-s + 6·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 1.94·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.79·31-s + 0.348·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s − 0.304·43-s − 0.149·45-s − 0.875·47-s + 1.12·51-s + 0.274·53-s + 0.269·55-s − 0.264·57-s − 1.56·59-s + 0.768·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6292670333\)
\(L(\frac12)\) \(\approx\) \(0.6292670333\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
23 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.26758793574649, −13.05412005533632, −12.55821512986725, −11.76360661963388, −11.58670369032548, −11.08016526398004, −10.67508951823822, −10.21034607595694, −9.534953362790801, −9.122357703871619, −8.492844470397780, −8.088341626989041, −7.568487153814808, −6.827043966188560, −6.643076383535375, −6.016426200046950, −5.375440071438322, −4.884671257207685, −4.402836314368264, −3.843702935322508, −3.211122999730424, −2.520626929056757, −1.891605890596319, −1.127235805319796, −0.2652856195865405, 0.2652856195865405, 1.127235805319796, 1.891605890596319, 2.520626929056757, 3.211122999730424, 3.843702935322508, 4.402836314368264, 4.884671257207685, 5.375440071438322, 6.016426200046950, 6.643076383535375, 6.827043966188560, 7.568487153814808, 8.088341626989041, 8.492844470397780, 9.122357703871619, 9.534953362790801, 10.21034607595694, 10.67508951823822, 11.08016526398004, 11.58670369032548, 11.76360661963388, 12.55821512986725, 13.05412005533632, 13.26758793574649

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