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Properties

Label 2-135240-1.1-c1-0-0
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 + 11-s − 15-s − 3·17-s − 4·19-s + 23-s + 25-s − 27-s − 7·29-s − 8·31-s − 33-s − 6·37-s − 10·41-s − 8·43-s + 45-s + 7·47-s + 3·51-s + 12·53-s + 55-s + 4·57-s + 6·61-s + 8·67-s − 69-s − 15·71-s + 7·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-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 − 1.29·29-s − 1.43·31-s − 0.174·33-s − 0.986·37-s − 1.56·41-s − 1.21·43-s + 0.149·45-s + 1.02·47-s + 0.420·51-s + 1.64·53-s + 0.134·55-s + 0.529·57-s + 0.768·61-s + 0.977·67-s − 0.120·69-s − 1.78·71-s + 0.819·73-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

\(L(1)\) \(\approx\) \(0.5422559716\)
\(L(\frac12)\) \(\approx\) \(0.5422559716\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.40705369685417, −12.94049125680819, −12.59255310957550, −11.91030413294336, −11.53657652117878, −11.06476670512209, −10.49116215971911, −10.25765570180839, −9.572463800236414, −9.062966435100262, −8.650163419422111, −8.198907202016302, −7.269686475994516, −6.982257881837413, −6.607824956404613, −5.889191911979293, −5.407512003613364, −5.119582172133721, −4.222608182996225, −3.917591120335550, −3.238776075128825, −2.362384864573136, −1.855348739689731, −1.333833423944887, −0.2226629283280570, 0.2226629283280570, 1.333833423944887, 1.855348739689731, 2.362384864573136, 3.238776075128825, 3.917591120335550, 4.222608182996225, 5.119582172133721, 5.407512003613364, 5.889191911979293, 6.607824956404613, 6.982257881837413, 7.269686475994516, 8.198907202016302, 8.650163419422111, 9.062966435100262, 9.572463800236414, 10.25765570180839, 10.49116215971911, 11.06476670512209, 11.53657652117878, 11.91030413294336, 12.59255310957550, 12.94049125680819, 13.40705369685417

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