Properties

Label 2-141570-1.1-c1-0-51
Degree $2$
Conductor $141570$
Sign $1$
Analytic cond. $1130.44$
Root an. cond. $33.6220$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s + 13-s − 2·14-s + 16-s + 8·17-s + 6·19-s + 20-s − 6·23-s + 25-s + 26-s − 2·28-s − 4·29-s + 32-s + 8·34-s − 2·35-s − 2·37-s + 6·38-s + 40-s − 2·41-s + 4·43-s − 6·46-s − 3·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.94·17-s + 1.37·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s + 0.176·32-s + 1.37·34-s − 0.338·35-s − 0.328·37-s + 0.973·38-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.884·46-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141570\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1130.44\)
Root analytic conductor: \(33.6220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{141570} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 141570,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.143333006\)
\(L(\frac12)\) \(\approx\) \(5.143333006\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.55511115902300, −12.93674192706077, −12.40559624187784, −12.08907731178501, −11.70646225495494, −11.03539462227748, −10.50376525778672, −9.975515031493470, −9.589086629188112, −9.356238486799610, −8.377229244934904, −7.939730311127344, −7.510027054203594, −6.831223417931529, −6.472277156123206, −5.743215971696785, −5.419936085482400, −5.189769435066361, −4.096518050656935, −3.714503654594025, −3.272463434651813, −2.683369434829506, −1.989259411775321, −1.271804887625846, −0.6219828876461286, 0.6219828876461286, 1.271804887625846, 1.989259411775321, 2.683369434829506, 3.272463434651813, 3.714503654594025, 4.096518050656935, 5.189769435066361, 5.419936085482400, 5.743215971696785, 6.472277156123206, 6.831223417931529, 7.510027054203594, 7.939730311127344, 8.377229244934904, 9.356238486799610, 9.589086629188112, 9.975515031493470, 10.50376525778672, 11.03539462227748, 11.70646225495494, 12.08907731178501, 12.40559624187784, 12.93674192706077, 13.55511115902300

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