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Properties

Label 2-333270-1.1-c1-0-97
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

−12.78701549910419, −12.32885401560916, −11.88507483204749, −11.31535494570120, −10.99160120706535, −10.41432006136037, −10.03966056076634, −9.717802177388558, −9.134524774552134, −8.762017830891974, −8.243010687821934, −7.883835225053155, −7.357616612147912, −6.583586310085503, −6.360344207264899, −6.145710928555583, −5.295126625246559, −4.866860756771864, −4.238299321934095, −3.659350723720748, −3.021504650133450, −2.548742009113258, −2.060201562390306, −1.216456514960927, −0.8764812914394296, 0, 0.8764812914394296, 1.216456514960927, 2.060201562390306, 2.548742009113258, 3.021504650133450, 3.659350723720748, 4.238299321934095, 4.866860756771864, 5.295126625246559, 6.145710928555583, 6.360344207264899, 6.583586310085503, 7.357616612147912, 7.883835225053155, 8.243010687821934, 8.762017830891974, 9.134524774552134, 9.717802177388558, 10.03966056076634, 10.41432006136037, 10.99160120706535, 11.31535494570120, 11.88507483204749, 12.32885401560916, 12.78701549910419

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