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Properties

Label 2-135240-1.1-c1-0-17
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 + 2·13-s − 15-s − 4·17-s + 2·19-s − 23-s + 25-s − 27-s + 2·31-s + 2·33-s + 10·37-s − 2·39-s − 2·41-s + 6·43-s + 45-s + 2·47-s + 4·51-s + 6·53-s − 2·55-s − 2·57-s − 4·59-s − 14·61-s + 2·65-s − 10·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s − 0.970·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.359·31-s + 0.348·33-s + 1.64·37-s − 0.320·39-s − 0.312·41-s + 0.914·43-s + 0.149·45-s + 0.291·47-s + 0.560·51-s + 0.824·53-s − 0.269·55-s − 0.264·57-s − 0.520·59-s − 1.79·61-s + 0.248·65-s − 1.22·67-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

\(L(1)\) \(\approx\) \(1.713164546\)
\(L(\frac12)\) \(\approx\) \(1.713164546\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.51934726908930, −13.02533021743856, −12.50124028099866, −11.97855054128235, −11.50525070848396, −10.97957463991105, −10.61755298674210, −10.20012401626051, −9.534843806605544, −9.173397566071670, −8.640207116830925, −7.971847127180211, −7.572970744047525, −6.953871146169179, −6.425293989997956, −5.863969324767511, −5.658006878341268, −4.838656178050530, −4.431836316742977, −3.918194550527808, −2.992371722372629, −2.613106547883176, −1.833532894432636, −1.190720420460564, −0.4300946442853699, 0.4300946442853699, 1.190720420460564, 1.833532894432636, 2.613106547883176, 2.992371722372629, 3.918194550527808, 4.431836316742977, 4.838656178050530, 5.658006878341268, 5.863969324767511, 6.425293989997956, 6.953871146169179, 7.572970744047525, 7.971847127180211, 8.640207116830925, 9.173397566071670, 9.534843806605544, 10.20012401626051, 10.61755298674210, 10.97957463991105, 11.50525070848396, 11.97855054128235, 12.50124028099866, 13.02533021743856, 13.51934726908930

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