Properties

Label 2-333270-1.1-c1-0-114
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·13-s − 14-s + 16-s − 7·17-s + 7·19-s + 20-s + 25-s + 4·26-s − 28-s + 2·29-s + 32-s − 7·34-s − 35-s − 6·37-s + 7·38-s + 40-s − 10·41-s − 43-s + 4·47-s + 49-s + 50-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 + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s + 1.60·19-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.371·29-s + 0.176·32-s − 1.20·34-s − 0.169·35-s − 0.986·37-s + 1.13·38-s + 0.158·40-s − 1.56·41-s − 0.152·43-s + 0.583·47-s + 1/7·49-s + 0.141·50-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.03618212755996, −12.30250505192933, −11.99839717413454, −11.50169180537315, −11.04479975736022, −10.68235674533361, −10.05992653985230, −9.805984057757060, −9.037813462846906, −8.740123855320090, −8.374417423736268, −7.553214873083097, −7.144635406606650, −6.633465585911251, −6.330655139343359, −5.789261389832259, −5.187401106307003, −4.965090278999596, −4.200205869540747, −3.672337748350423, −3.347909344669658, −2.632595598744500, −2.183756630755403, −1.481738277457630, −0.9467831919214784, 0, 0.9467831919214784, 1.481738277457630, 2.183756630755403, 2.632595598744500, 3.347909344669658, 3.672337748350423, 4.200205869540747, 4.965090278999596, 5.187401106307003, 5.789261389832259, 6.330655139343359, 6.633465585911251, 7.144635406606650, 7.553214873083097, 8.374417423736268, 8.740123855320090, 9.037813462846906, 9.805984057757060, 10.05992653985230, 10.68235674533361, 11.04479975736022, 11.50169180537315, 11.99839717413454, 12.30250505192933, 13.03618212755996

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