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Properties

Label 2-135240-1.1-c1-0-19
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 + 4·13-s − 15-s − 3·17-s + 4·19-s − 23-s + 25-s − 27-s + 5·29-s − 31-s + 2·33-s + 3·37-s − 4·39-s − 2·41-s + 43-s + 45-s − 4·47-s + 3·51-s − 3·53-s − 2·55-s − 4·57-s − 9·59-s + 14·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.727·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s − 0.179·31-s + 0.348·33-s + 0.493·37-s − 0.640·39-s − 0.312·41-s + 0.152·43-s + 0.149·45-s − 0.583·47-s + 0.420·51-s − 0.412·53-s − 0.269·55-s − 0.529·57-s − 1.17·59-s + 1.79·61-s + 0.496·65-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

\(L(1)\) \(\approx\) \(1.847419436\)
\(L(\frac12)\) \(\approx\) \(1.847419436\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.29801603077816, −13.07439755795712, −12.57933133410431, −11.89892773758137, −11.42749127414052, −11.18229513434812, −10.48381745521176, −10.15275769928953, −9.737338108800871, −8.965699915118734, −8.690632859308929, −8.074355846432651, −7.477437627797911, −6.996815219140259, −6.376941454650142, −5.961891060368818, −5.567003500640389, −4.866373590821684, −4.492553902396493, −3.790899021102523, −3.095155330259008, −2.614122320554490, −1.735632541377954, −1.254281569585545, −0.4410369481360072, 0.4410369481360072, 1.254281569585545, 1.735632541377954, 2.614122320554490, 3.095155330259008, 3.790899021102523, 4.492553902396493, 4.866373590821684, 5.567003500640389, 5.961891060368818, 6.376941454650142, 6.996815219140259, 7.477437627797911, 8.074355846432651, 8.690632859308929, 8.965699915118734, 9.737338108800871, 10.15275769928953, 10.48381745521176, 11.18229513434812, 11.42749127414052, 11.89892773758137, 12.57933133410431, 13.07439755795712, 13.29801603077816

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