Properties

Label 2-159120-1.1-c1-0-19
Degree $2$
Conductor $159120$
Sign $1$
Analytic cond. $1270.57$
Root an. cond. $35.6451$
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  + 5-s + 4·11-s + 13-s − 17-s + 8·19-s − 8·23-s + 25-s − 6·29-s − 6·37-s − 6·41-s − 4·43-s + 4·47-s − 7·49-s + 10·53-s + 4·55-s + 6·61-s + 65-s − 2·73-s − 8·79-s − 85-s − 2·89-s + 8·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 49-s + 1.37·53-s + 0.539·55-s + 0.768·61-s + 0.124·65-s − 0.234·73-s − 0.900·79-s − 0.108·85-s − 0.211·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1270.57\)
Root analytic conductor: \(35.6451\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 159120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.789301863\)
\(L(\frac12)\) \(\approx\) \(2.789301863\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.29858016079491, −12.94405420927586, −12.11743219126463, −11.86258854380268, −11.53961795973821, −10.99954065100653, −10.19181128271876, −9.979488982211592, −9.531502267868028, −8.872155092203331, −8.683126538622110, −7.900452303802443, −7.433048940734333, −6.894862375620851, −6.453289508689976, −5.811139464329768, −5.495804737222022, −4.881420136617151, −4.179474321800294, −3.587352892592465, −3.348581294588116, −2.395243579461357, −1.758495228707046, −1.354818812667235, −0.4892914516740358, 0.4892914516740358, 1.354818812667235, 1.758495228707046, 2.395243579461357, 3.348581294588116, 3.587352892592465, 4.179474321800294, 4.881420136617151, 5.495804737222022, 5.811139464329768, 6.453289508689976, 6.894862375620851, 7.433048940734333, 7.900452303802443, 8.683126538622110, 8.872155092203331, 9.531502267868028, 9.979488982211592, 10.19181128271876, 10.99954065100653, 11.53961795973821, 11.86258854380268, 12.11743219126463, 12.94405420927586, 13.29858016079491

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