↑

Properties

Label 2-333270-1.1-c1-0-105
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 + 3·11-s − 5·13-s + 14-s + 16-s − 3·19-s − 20-s + 3·22-s + 25-s − 5·26-s + 28-s + 10·29-s + 5·31-s + 32-s − 35-s + 12·37-s − 3·38-s − 40-s − 6·41-s + 3·43-s + 3·44-s − 8·47-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 + 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.688·19-s − 0.223·20-s + 0.639·22-s + 1/5·25-s − 0.980·26-s + 0.188·28-s + 1.85·29-s + 0.898·31-s + 0.176·32-s − 0.169·35-s + 1.97·37-s − 0.486·38-s − 0.158·40-s − 0.937·41-s + 0.457·43-s + 0.452·44-s − 1.16·47-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−12.72593476701096, −12.27859559087204, −11.97664773414708, −11.60092262319853, −11.19213107366251, −10.62894817834998, −10.00768206804464, −9.857748705876350, −9.164021887856490, −8.577432201760449, −8.126804372217754, −7.808289010362324, −7.100560295081498, −6.715261885404395, −6.414430726784520, −5.753729100973307, −5.158111846667387, −4.637244930184356, −4.365232967546160, −3.925300963888550, −3.176766499727683, −2.579403777745949, −2.340610636462303, −1.388535462519946, −0.9160274571399132, 0, 0.9160274571399132, 1.388535462519946, 2.340610636462303, 2.579403777745949, 3.176766499727683, 3.925300963888550, 4.365232967546160, 4.637244930184356, 5.158111846667387, 5.753729100973307, 6.414430726784520, 6.715261885404395, 7.100560295081498, 7.808289010362324, 8.126804372217754, 8.577432201760449, 9.164021887856490, 9.857748705876350, 10.00768206804464, 10.62894817834998, 11.19213107366251, 11.60092262319853, 11.97664773414708, 12.27859559087204, 12.72593476701096

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