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Properties

Label 2-333270-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 6·11-s − 14-s + 16-s + 2·17-s − 20-s − 6·22-s + 25-s − 28-s + 6·29-s + 10·31-s + 32-s + 2·34-s + 35-s − 6·37-s − 40-s − 6·41-s − 10·43-s − 6·44-s − 6·47-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 − 1.80·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.188·28-s + 1.11·29-s + 1.79·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.986·37-s − 0.158·40-s − 0.937·41-s − 1.52·43-s − 0.904·44-s − 0.875·47-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: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.091110545\)
\(L(\frac12)\) \(\approx\) \(2.091110545\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.50418863617189, −12.18776211391065, −11.87435809693358, −11.29463515909571, −10.79851026514454, −10.22378048648735, −10.10494878088063, −9.656992998251953, −8.628760013009648, −8.395785456592383, −8.029516756254667, −7.452479663499261, −6.985171642439702, −6.486794767344662, −6.073428341463270, −5.348975304499602, −4.976624139016194, −4.741997801751725, −3.976341189916859, −3.337979134588319, −3.044061420155241, −2.529684474637763, −1.924568961453393, −1.087316316236382, −0.3506748742944074, 0.3506748742944074, 1.087316316236382, 1.924568961453393, 2.529684474637763, 3.044061420155241, 3.337979134588319, 3.976341189916859, 4.741997801751725, 4.976624139016194, 5.348975304499602, 6.073428341463270, 6.486794767344662, 6.985171642439702, 7.452479663499261, 8.029516756254667, 8.395785456592383, 8.628760013009648, 9.656992998251953, 10.10494878088063, 10.22378048648735, 10.79851026514454, 11.29463515909571, 11.87435809693358, 12.18776211391065, 12.50418863617189

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