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Properties

Label 2-135240-1.1-c1-0-5
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 + 13-s − 15-s − 4·17-s − 7·19-s + 23-s + 25-s − 27-s − 7·29-s + 9·31-s − 2·33-s − 39-s − 10·43-s + 45-s − 8·47-s + 4·51-s + 3·53-s + 2·55-s + 7·57-s + 5·59-s − 7·61-s + 65-s − 14·67-s − 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 1.60·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.29·29-s + 1.61·31-s − 0.348·33-s − 0.160·39-s − 1.52·43-s + 0.149·45-s − 1.16·47-s + 0.560·51-s + 0.412·53-s + 0.269·55-s + 0.927·57-s + 0.650·59-s − 0.896·61-s + 0.124·65-s − 1.71·67-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

\(L(1)\) \(\approx\) \(0.9350110672\)
\(L(\frac12)\) \(\approx\) \(0.9350110672\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
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

−13.45447066394712, −13.16811345480949, −12.33042059143408, −12.11045053677277, −11.42858649710272, −11.06482929659839, −10.63985191012232, −10.10315442096429, −9.623985737092878, −9.102224902846626, −8.531075531655146, −8.254368899309304, −7.439553774697653, −6.809769203380315, −6.425237177752399, −6.182115444256082, −5.448927259704756, −4.892308923288342, −4.322541885208413, −3.965011871076124, −3.142018152589839, −2.463910751882513, −1.773725243907536, −1.343159204427273, −0.2959522658990370, 0.2959522658990370, 1.343159204427273, 1.773725243907536, 2.463910751882513, 3.142018152589839, 3.965011871076124, 4.322541885208413, 4.892308923288342, 5.448927259704756, 6.182115444256082, 6.425237177752399, 6.809769203380315, 7.439553774697653, 8.254368899309304, 8.531075531655146, 9.102224902846626, 9.623985737092878, 10.10315442096429, 10.63985191012232, 11.06482929659839, 11.42858649710272, 12.11045053677277, 12.33042059143408, 13.16811345480949, 13.45447066394712

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