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Properties

Label 2-135240-1.1-c1-0-32
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 − 15-s − 7·17-s − 4·19-s + 23-s + 25-s − 27-s + 9·29-s + 9·31-s + 2·33-s + 3·37-s − 2·41-s − 43-s + 45-s + 8·47-s + 7·51-s + 9·53-s − 2·55-s + 4·57-s + 13·59-s − 2·61-s + 16·67-s − 69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.258·15-s − 1.69·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 1.61·31-s + 0.348·33-s + 0.493·37-s − 0.312·41-s − 0.152·43-s + 0.149·45-s + 1.16·47-s + 0.980·51-s + 1.23·53-s − 0.269·55-s + 0.529·57-s + 1.69·59-s − 0.256·61-s + 1.95·67-s − 0.120·69-s − 0.593·71-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

\(L(1)\) \(\approx\) \(2.037568076\)
\(L(\frac12)\) \(\approx\) \(2.037568076\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.32528721866358, −12.97251436826656, −12.63574925850562, −11.79498200514218, −11.68375922011136, −10.95712746176480, −10.54521293452773, −10.16198497479825, −9.767027201431869, −8.973721292087195, −8.505024942730983, −8.306206377324069, −7.406810351447750, −6.882300863632449, −6.458385314539190, −6.113525763723797, −5.408242831773188, −4.874205240834621, −4.419943259687604, −3.990304527592979, −3.001553176115109, −2.378817681678059, −2.123747860204861, −1.021817238465977, −0.5110543728525842, 0.5110543728525842, 1.021817238465977, 2.123747860204861, 2.378817681678059, 3.001553176115109, 3.990304527592979, 4.419943259687604, 4.874205240834621, 5.408242831773188, 6.113525763723797, 6.458385314539190, 6.882300863632449, 7.406810351447750, 8.306206377324069, 8.505024942730983, 8.973721292087195, 9.767027201431869, 10.16198497479825, 10.54521293452773, 10.95712746176480, 11.68375922011136, 11.79498200514218, 12.63574925850562, 12.97251436826656, 13.32528721866358

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