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Properties

Label 2-135240-1.1-c1-0-11
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 + 15-s + 6·17-s − 4·19-s + 23-s + 25-s − 27-s − 29-s + 7·31-s − 4·33-s + 7·37-s + 9·41-s − 9·43-s − 45-s − 8·47-s − 6·51-s − 14·53-s − 4·55-s + 4·57-s − 4·59-s − 5·67-s − 69-s − 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s + 1.25·31-s − 0.696·33-s + 1.15·37-s + 1.40·41-s − 1.37·43-s − 0.149·45-s − 1.16·47-s − 0.840·51-s − 1.92·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s − 0.610·67-s − 0.120·69-s − 0.949·71-s + 1.63·73-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

\(L(1)\) \(\approx\) \(1.554820322\)
\(L(\frac12)\) \(\approx\) \(1.554820322\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.38777695357933, −12.72595289595035, −12.51821785520508, −11.96670717897735, −11.50236250324157, −11.20201258236015, −10.64702786224893, −10.00350208898008, −9.635112030052766, −9.215693771123722, −8.441575926601156, −8.033875377764632, −7.645220750719395, −6.837509044653394, −6.540590542230939, −6.049726451005031, −5.489966421338521, −4.834336013602735, −4.243891139184387, −3.984059622069524, −3.104717192427586, −2.753939592636275, −1.512431390334675, −1.343806690586848, −0.4119480592043541, 0.4119480592043541, 1.343806690586848, 1.512431390334675, 2.753939592636275, 3.104717192427586, 3.984059622069524, 4.243891139184387, 4.834336013602735, 5.489966421338521, 6.049726451005031, 6.540590542230939, 6.837509044653394, 7.645220750719395, 8.033875377764632, 8.441575926601156, 9.215693771123722, 9.635112030052766, 10.00350208898008, 10.64702786224893, 11.20201258236015, 11.50236250324157, 11.96670717897735, 12.51821785520508, 12.72595289595035, 13.38777695357933

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