Properties

Label 2-129360-1.1-c1-0-125
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 − 6·13-s − 15-s + 6·17-s + 4·19-s + 25-s − 27-s − 2·29-s − 8·31-s − 33-s − 2·37-s + 6·39-s − 4·41-s − 6·43-s + 45-s + 8·47-s − 6·51-s − 2·53-s + 55-s − 4·57-s − 8·59-s + 2·61-s − 6·65-s − 8·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-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 + 0.960·39-s − 0.624·41-s − 0.914·43-s + 0.149·45-s + 1.16·47-s − 0.840·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s − 1.04·59-s + 0.256·61-s − 0.744·65-s − 0.977·67-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: \(1\)
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.85117699171661, −13.13384989043363, −12.64419833181608, −12.27767072316910, −11.86974131425262, −11.43355558649739, −10.82592589393066, −10.19117774612107, −9.927352213248916, −9.511847026893835, −9.010020011492485, −8.333689167783248, −7.608446220421973, −7.241895904968489, −7.009707713971030, −6.108191429511960, −5.600009014297826, −5.348880473068465, −4.747258812037943, −4.192395520064745, −3.261767908972969, −3.083870372192797, −2.043050334994597, −1.631890207455675, −0.8080962624552271, 0, 0.8080962624552271, 1.631890207455675, 2.043050334994597, 3.083870372192797, 3.261767908972969, 4.192395520064745, 4.747258812037943, 5.348880473068465, 5.600009014297826, 6.108191429511960, 7.009707713971030, 7.241895904968489, 7.608446220421973, 8.333689167783248, 9.010020011492485, 9.511847026893835, 9.927352213248916, 10.19117774612107, 10.82592589393066, 11.43355558649739, 11.86974131425262, 12.27767072316910, 12.64419833181608, 13.13384989043363, 13.85117699171661

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