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Properties

Label 2-333270-1.1-c1-0-119
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 − 3·11-s + 5·13-s − 14-s + 16-s + 8·17-s − 19-s + 20-s + 3·22-s + 25-s − 5·26-s + 28-s − 2·29-s − 7·31-s − 32-s − 8·34-s + 35-s + 38-s − 40-s + 10·41-s + 7·43-s − 3·44-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 − 0.904·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s + 1/5·25-s − 0.980·26-s + 0.188·28-s − 0.371·29-s − 1.25·31-s − 0.176·32-s − 1.37·34-s + 0.169·35-s + 0.162·38-s − 0.158·40-s + 1.56·41-s + 1.06·43-s − 0.452·44-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.75089599504731, −12.42922430193574, −11.85023825145269, −11.19958415493983, −10.97802659198746, −10.54147742335193, −10.14665384082819, −9.528671630830539, −9.275759785943417, −8.668956384734404, −8.197947630794682, −7.821690880394288, −7.434554706497935, −6.927498606557909, −6.182055471800224, −5.751124674685069, −5.550087857939438, −4.959296206148899, −4.134962192119127, −3.648172192152086, −3.138393580996618, −2.489356142918334, −1.984207910769977, −1.203390681426116, −0.9856683046138208, 0, 0.9856683046138208, 1.203390681426116, 1.984207910769977, 2.489356142918334, 3.138393580996618, 3.648172192152086, 4.134962192119127, 4.959296206148899, 5.550087857939438, 5.751124674685069, 6.182055471800224, 6.927498606557909, 7.434554706497935, 7.821690880394288, 8.197947630794682, 8.668956384734404, 9.275759785943417, 9.528671630830539, 10.14665384082819, 10.54147742335193, 10.97802659198746, 11.19958415493983, 11.85023825145269, 12.42922430193574, 12.75089599504731

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