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Properties

Label 2-129360-1.1-c1-0-120
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 − 6·17-s − 4·19-s + 25-s + 27-s + 6·29-s − 4·31-s + 33-s + 2·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s − 6·51-s + 6·53-s − 55-s − 4·57-s − 2·61-s + 2·65-s + 4·67-s − 2·73-s + 75-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 − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.840·51-s + 0.824·53-s − 0.134·55-s − 0.529·57-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.234·73-s + 0.115·75-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.72161255720844, −13.27209507735120, −12.79003078973900, −12.32867495458138, −11.87501915378140, −11.28887359713792, −10.83363273050193, −10.39354969478451, −9.790404157861713, −9.305023709611799, −8.657176845488592, −8.551845769721196, −7.907883748462639, −7.250151995693388, −6.897649519398832, −6.413848442921516, −5.793632726109068, −4.997553446441963, −4.485712694766962, −4.145440628254601, −3.490973125945882, −2.825336328741782, −2.258000462185143, −1.766411757484980, −0.7910421673097111, 0, 0.7910421673097111, 1.766411757484980, 2.258000462185143, 2.825336328741782, 3.490973125945882, 4.145440628254601, 4.485712694766962, 4.997553446441963, 5.793632726109068, 6.413848442921516, 6.897649519398832, 7.250151995693388, 7.907883748462639, 8.551845769721196, 8.657176845488592, 9.305023709611799, 9.790404157861713, 10.39354969478451, 10.83363273050193, 11.28887359713792, 11.87501915378140, 12.32867495458138, 12.79003078973900, 13.27209507735120, 13.72161255720844

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