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Properties

Label 2-135240-1.1-c1-0-24
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 − 4·11-s − 2·13-s + 15-s + 2·17-s + 4·19-s − 23-s + 25-s + 27-s + 6·29-s − 4·33-s − 10·37-s − 2·39-s + 2·41-s + 4·43-s + 45-s − 8·47-s + 2·51-s + 6·53-s − 4·55-s + 4·57-s − 10·61-s − 2·65-s + 4·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.696·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.120·69-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

\(L(1)\) \(\approx\) \(2.699323483\)
\(L(\frac12)\) \(\approx\) \(2.699323483\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.41528810255010, −13.12443470110813, −12.43902839083471, −12.17538708856663, −11.60481293607166, −10.91676204956346, −10.38892289531110, −10.01855267822399, −9.748391859166202, −8.973056689136685, −8.645204529962132, −8.041364253403859, −7.486645000497801, −7.286812630846495, −6.509671683641400, −5.961977349408563, −5.272426781063878, −5.031973303772265, −4.390042979837281, −3.609546184705539, −2.991466224337106, −2.692501053426790, −1.959141286648110, −1.345346763892541, −0.4658268852184636, 0.4658268852184636, 1.345346763892541, 1.959141286648110, 2.692501053426790, 2.991466224337106, 3.609546184705539, 4.390042979837281, 5.031973303772265, 5.272426781063878, 5.961977349408563, 6.509671683641400, 7.286812630846495, 7.486645000497801, 8.041364253403859, 8.645204529962132, 8.973056689136685, 9.748391859166202, 10.01855267822399, 10.38892289531110, 10.91676204956346, 11.60481293607166, 12.17538708856663, 12.43902839083471, 13.12443470110813, 13.41528810255010

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