Properties

Label 2-333270-1.1-c1-0-1
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 4·11-s − 6·13-s + 14-s + 16-s + 4·17-s − 6·19-s + 20-s + 4·22-s + 25-s + 6·26-s − 28-s − 5·29-s − 3·31-s − 32-s − 4·34-s − 35-s − 2·37-s + 6·38-s − 40-s + 3·41-s + 8·43-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 − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 0.928·29-s − 0.538·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s − 0.328·37-s + 0.973·38-s − 0.158·40-s + 0.468·41-s + 1.21·43-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: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3427768241\)
\(L(\frac12)\) \(\approx\) \(0.3427768241\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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

−12.51469883111320, −12.34083843047571, −11.59509838056872, −11.08294101370960, −10.65986634937461, −10.19754571127657, −9.895871337324590, −9.506091023622970, −8.995157926553199, −8.495664232678868, −7.928016246486151, −7.512040165963620, −7.218030393477282, −6.652648679728701, −6.009287317519558, −5.602520271051789, −5.183129301727590, −4.610093625941358, −3.976674214348894, −3.268973597206701, −2.721561520654363, −2.240177031929225, −1.922012572078231, −0.9747138992129462, −0.1874130925901766, 0.1874130925901766, 0.9747138992129462, 1.922012572078231, 2.240177031929225, 2.721561520654363, 3.268973597206701, 3.976674214348894, 4.610093625941358, 5.183129301727590, 5.602520271051789, 6.009287317519558, 6.652648679728701, 7.218030393477282, 7.512040165963620, 7.928016246486151, 8.495664232678868, 8.995157926553199, 9.506091023622970, 9.895871337324590, 10.19754571127657, 10.65986634937461, 11.08294101370960, 11.59509838056872, 12.34083843047571, 12.51469883111320

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