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Properties

Label 2-333270-1.1-c1-0-87
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 + 2·13-s − 14-s + 16-s − 2·19-s − 20-s + 25-s + 2·26-s − 28-s − 6·29-s + 8·31-s + 32-s + 35-s + 4·37-s − 2·38-s − 40-s − 6·41-s − 2·43-s + 6·47-s + 49-s + 50-s + 2·52-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 + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.169·35-s + 0.657·37-s − 0.324·38-s − 0.158·40-s − 0.937·41-s − 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.141·50-s + 0.277·52-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.80626617778596, −12.39331346973342, −11.90303245336675, −11.63488519518996, −11.03453908632452, −10.61816405767390, −10.27912475496105, −9.655063451498058, −9.103499877709033, −8.740042473408407, −8.087325059832471, −7.718092733614179, −7.266165244735687, −6.574106439660863, −6.334411552736113, −5.840694647444000, −5.257567495746324, −4.703132256597602, −4.250095746071219, −3.758300335744121, −3.281428856929121, −2.765679123803489, −2.160110098932904, −1.493823804081381, −0.8044219147980082, 0, 0.8044219147980082, 1.493823804081381, 2.160110098932904, 2.765679123803489, 3.281428856929121, 3.758300335744121, 4.250095746071219, 4.703132256597602, 5.257567495746324, 5.840694647444000, 6.334411552736113, 6.574106439660863, 7.266165244735687, 7.718092733614179, 8.087325059832471, 8.740042473408407, 9.103499877709033, 9.655063451498058, 10.27912475496105, 10.61816405767390, 11.03453908632452, 11.63488519518996, 11.90303245336675, 12.39331346973342, 12.80626617778596

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