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Properties

Label 2-129360-1.1-c1-0-111
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 $0$

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 + 8·19-s + 6·23-s + 25-s + 27-s + 6·29-s + 2·31-s + 33-s + 2·37-s − 2·39-s − 8·43-s + 45-s − 12·47-s + 6·51-s + 6·53-s + 55-s + 8·57-s + 6·59-s − 8·61-s − 2·65-s − 2·67-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 + 1.83·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s + 0.134·55-s + 1.05·57-s + 0.781·59-s − 1.02·61-s − 0.248·65-s − 0.244·67-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: \(0\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.262524776\)
\(L(\frac12)\) \(\approx\) \(5.262524776\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
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

−13.53172672843550, −13.17643857718694, −12.45966972899032, −12.09779918502600, −11.67359689318608, −11.13265749719819, −10.39619800659046, −9.950975513062318, −9.645867217806937, −9.258957733518704, −8.571738781512043, −8.069948627335927, −7.670070659037287, −6.976746763088872, −6.761894614883661, −5.947551551924928, −5.326744345489767, −5.014180486369211, −4.422391073529918, −3.509814098668904, −3.111692428172692, −2.790449078165450, −1.855107589359461, −1.223158283145512, −0.7396908571337522, 0.7396908571337522, 1.223158283145512, 1.855107589359461, 2.790449078165450, 3.111692428172692, 3.509814098668904, 4.422391073529918, 5.014180486369211, 5.326744345489767, 5.947551551924928, 6.761894614883661, 6.976746763088872, 7.670070659037287, 8.069948627335927, 8.571738781512043, 9.258957733518704, 9.645867217806937, 9.950975513062318, 10.39619800659046, 11.13265749719819, 11.67359689318608, 12.09779918502600, 12.45966972899032, 13.17643857718694, 13.53172672843550

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