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Properties

Label 2-333270-1.1-c1-0-107
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 + 4·11-s + 2·13-s + 14-s + 16-s − 6·17-s + 4·19-s − 20-s + 4·22-s + 25-s + 2·26-s + 28-s − 2·29-s − 8·31-s + 32-s − 6·34-s − 35-s − 6·37-s + 4·38-s − 40-s + 6·41-s + 4·43-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 + 1.20·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s − 0.986·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 0.609·43-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.86110063094386, −12.39682196206196, −11.78997044537020, −11.56460817376925, −11.03349726950041, −10.89783326414608, −10.21694404195433, −9.467396045831564, −9.180568715832138, −8.720912975197420, −8.229734003055413, −7.626513352344112, −7.135109444437048, −6.784832114590036, −6.354621458971517, −5.601252675713931, −5.423110108341151, −4.630552748414116, −4.244992660719715, −3.745875210971133, −3.463738990297381, −2.659965530808240, −2.068089311490337, −1.502443300126504, −0.9242491562533194, 0, 0.9242491562533194, 1.502443300126504, 2.068089311490337, 2.659965530808240, 3.463738990297381, 3.745875210971133, 4.244992660719715, 4.630552748414116, 5.423110108341151, 5.601252675713931, 6.354621458971517, 6.784832114590036, 7.135109444437048, 7.626513352344112, 8.229734003055413, 8.720912975197420, 9.180568715832138, 9.467396045831564, 10.21694404195433, 10.89783326414608, 11.03349726950041, 11.56460817376925, 11.78997044537020, 12.39682196206196, 12.86110063094386

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