Properties

Label 2-333270-1.1-c1-0-101
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

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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 + 20-s + 25-s + 4·26-s − 28-s − 5·29-s − 7·31-s + 32-s − 35-s − 6·37-s + 40-s − 3·41-s + 6·43-s + 4·47-s + 49-s + 50-s + 4·52-s − 4·53-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 + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.928·29-s − 1.25·31-s + 0.176·32-s − 0.169·35-s − 0.986·37-s + 0.158·40-s − 0.468·41-s + 0.914·43-s + 0.583·47-s + 1/7·49-s + 0.141·50-s + 0.554·52-s − 0.549·53-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: \(1\)
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 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90039340652195, −12.58657477963029, −11.94835185812234, −11.34283443867483, −11.19948048297472, −10.57151205907264, −10.20388946471835, −9.678036459635079, −9.166772850007789, −8.616273857156650, −8.419586727983954, −7.471357551675419, −7.228240385058079, −6.770189284618236, −6.069940334299621, −5.798393106812260, −5.453805991528681, −4.796345925561021, −4.190539794232558, −3.690828310473801, −3.342213115141495, −2.714341946251083, −2.022386454775732, −1.615405858251395, −0.8966909328328045, 0, 0.8966909328328045, 1.615405858251395, 2.022386454775732, 2.714341946251083, 3.342213115141495, 3.690828310473801, 4.190539794232558, 4.796345925561021, 5.453805991528681, 5.798393106812260, 6.069940334299621, 6.770189284618236, 7.228240385058079, 7.471357551675419, 8.419586727983954, 8.616273857156650, 9.166772850007789, 9.678036459635079, 10.20388946471835, 10.57151205907264, 11.19948048297472, 11.34283443867483, 11.94835185812234, 12.58657477963029, 12.90039340652195

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