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Properties

Label 2-129360-1.1-c1-0-117
Degree $2$
Conductor $129360$
Sign $-1$
Analytic cond. $1032.94$
Root an. cond. $32.1394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 2·17-s + 6·19-s + 25-s − 27-s + 2·29-s − 4·31-s + 33-s − 2·37-s + 2·39-s − 4·43-s − 45-s − 8·47-s − 2·51-s − 4·53-s + 55-s − 6·57-s + 14·59-s + 14·61-s + 2·65-s + 2·67-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 1.37·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.320·39-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 0.280·51-s − 0.549·53-s + 0.134·55-s − 0.794·57-s + 1.82·59-s + 1.79·61-s + 0.248·65-s + 0.244·67-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1032.94\)
Root analytic conductor: \(32.1394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.70407774361884, −13.11915118021753, −12.75651642608460, −12.19773595061332, −11.74459791263449, −11.47051572856135, −10.91102121465954, −10.34117247143349, −9.801402048797466, −9.614616937467007, −8.811390566281293, −8.286718179880234, −7.707728187761417, −7.403177857557365, −6.737899879266245, −6.412556834153868, −5.471402663613640, −5.225041311585449, −4.874699415437679, −3.945326254890301, −3.598120474502591, −2.914570627654662, −2.257886543403928, −1.429653492396200, −0.7773401620617030, 0, 0.7773401620617030, 1.429653492396200, 2.257886543403928, 2.914570627654662, 3.598120474502591, 3.945326254890301, 4.874699415437679, 5.225041311585449, 5.471402663613640, 6.412556834153868, 6.737899879266245, 7.403177857557365, 7.707728187761417, 8.286718179880234, 8.811390566281293, 9.614616937467007, 9.801402048797466, 10.34117247143349, 10.91102121465954, 11.47051572856135, 11.74459791263449, 12.19773595061332, 12.75651642608460, 13.11915118021753, 13.70407774361884

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