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Properties

Label 2-129360-1.1-c1-0-109
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 − 8·19-s − 8·23-s + 25-s − 27-s − 2·29-s + 8·31-s + 33-s − 2·37-s + 2·39-s + 6·41-s + 12·43-s + 45-s − 2·51-s − 6·53-s − 55-s + 8·57-s − 10·61-s − 2·65-s + 16·67-s + 8·69-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 − 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.149·45-s − 0.280·51-s − 0.824·53-s − 0.134·55-s + 1.05·57-s − 1.28·61-s − 0.248·65-s + 1.95·67-s + 0.963·69-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 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.83001581466811, −13.13437136499001, −12.65627506149307, −12.33489941856114, −11.94929509571994, −11.24974208480989, −10.76300320884287, −10.39277723908445, −9.960235757390094, −9.435695948273584, −8.963333408387656, −8.173763990031345, −7.912056510147902, −7.352829140754316, −6.544088133434386, −6.291720586476120, −5.796890630439132, −5.275432058938903, −4.540026407862022, −4.255889939096572, −3.592828574676677, −2.630962769285497, −2.284051809271889, −1.625948360958247, −0.7357084577528127, 0, 0.7357084577528127, 1.625948360958247, 2.284051809271889, 2.630962769285497, 3.592828574676677, 4.255889939096572, 4.540026407862022, 5.275432058938903, 5.796890630439132, 6.291720586476120, 6.544088133434386, 7.352829140754316, 7.912056510147902, 8.173763990031345, 8.963333408387656, 9.435695948273584, 9.960235757390094, 10.39277723908445, 10.76300320884287, 11.24974208480989, 11.94929509571994, 12.33489941856114, 12.65627506149307, 13.13437136499001, 13.83001581466811

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