Properties

Label 2-333270-1.1-c1-0-132
Degree $2$
Conductor $333270$
Sign $1$
Analytic cond. $2661.17$
Root an. cond. $51.5865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 14-s + 16-s − 4·17-s − 6·19-s − 20-s + 25-s − 28-s − 8·31-s + 32-s − 4·34-s + 35-s + 6·37-s − 6·38-s − 40-s − 6·41-s + 8·43-s − 12·47-s + 49-s + 50-s − 6·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.223·20-s + 1/5·25-s − 0.188·28-s − 1.43·31-s + 0.176·32-s − 0.685·34-s + 0.169·35-s + 0.986·37-s − 0.973·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.141·50-s − 0.824·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2661.17\)
Root analytic conductor: \(51.5865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{333270} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−12.92807828398354, −12.67069047174188, −12.32010051821985, −11.68535794781981, −11.19874604501877, −10.86540358385100, −10.61827055172671, −9.812955602523174, −9.436930065436532, −8.934158923245353, −8.368548285294291, −7.990124945097192, −7.441175738804772, −6.806969695448628, −6.608449747118250, −6.056709602931868, −5.559604619468508, −4.935301938796349, −4.442503554898902, −4.057897099654108, −3.592089416620302, −2.912233180460881, −2.499407169100284, −1.817700638923464, −1.283779347256611, 0, 0, 1.283779347256611, 1.817700638923464, 2.499407169100284, 2.912233180460881, 3.592089416620302, 4.057897099654108, 4.442503554898902, 4.935301938796349, 5.559604619468508, 6.056709602931868, 6.608449747118250, 6.806969695448628, 7.441175738804772, 7.990124945097192, 8.368548285294291, 8.934158923245353, 9.436930065436532, 9.812955602523174, 10.61827055172671, 10.86540358385100, 11.19874604501877, 11.68535794781981, 12.32010051821985, 12.67069047174188, 12.92807828398354

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