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Properties

Label 2-129360-1.1-c1-0-41
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 + 15-s − 4·17-s − 2·23-s + 25-s + 27-s + 6·29-s − 8·31-s + 33-s − 8·37-s − 2·41-s + 4·43-s + 45-s + 12·47-s − 4·51-s + 14·53-s + 55-s − 10·61-s − 2·67-s − 2·69-s + 8·71-s − 6·73-s + 75-s − 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.258·15-s − 0.970·17-s − 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s − 1.31·37-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.75·47-s − 0.560·51-s + 1.92·53-s + 0.134·55-s − 1.28·61-s − 0.244·67-s − 0.240·69-s + 0.949·71-s − 0.702·73-s + 0.115·75-s − 0.900·79-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

\(L(1)\) \(\approx\) \(3.039207772\)
\(L(\frac12)\) \(\approx\) \(3.039207772\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.55800035446069, −13.12659208101234, −12.51854575370069, −12.14754887801907, −11.64662894004791, −10.94715226076351, −10.48959353602946, −10.21721645177484, −9.463204867299682, −9.032521021188487, −8.734722314094541, −8.241075332937693, −7.473508785918390, −7.091995230061221, −6.650883424641261, −5.955826663459273, −5.527449973074453, −4.874729744088208, −4.201112346903859, −3.865822862912398, −3.102171019052585, −2.505336388202131, −1.984439430505935, −1.384303626188685, −0.4948460697797159, 0.4948460697797159, 1.384303626188685, 1.984439430505935, 2.505336388202131, 3.102171019052585, 3.865822862912398, 4.201112346903859, 4.874729744088208, 5.527449973074453, 5.955826663459273, 6.650883424641261, 7.091995230061221, 7.473508785918390, 8.241075332937693, 8.734722314094541, 9.032521021188487, 9.463204867299682, 10.21721645177484, 10.48959353602946, 10.94715226076351, 11.64662894004791, 12.14754887801907, 12.51854575370069, 13.12659208101234, 13.55800035446069

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