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Properties

Label 2-333270-1.1-c1-0-53
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 − 2·11-s − 6·13-s − 14-s + 16-s + 6·17-s − 20-s − 2·22-s + 25-s − 6·26-s − 28-s − 4·29-s − 2·31-s + 32-s + 6·34-s + 35-s − 4·37-s − 40-s − 2·41-s + 4·43-s − 2·44-s + 49-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 − 0.603·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.223·20-s − 0.426·22-s + 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.742·29-s − 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s − 0.657·37-s − 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.301·44-s + 1/7·49-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−12.68760937804116, −12.43897490049926, −12.04024433930334, −11.62792061624497, −11.08077437849526, −10.52141759701741, −10.14378482553263, −9.784408587952825, −9.193898288444800, −8.758936258251993, −7.884521920401409, −7.617783898184966, −7.445650051539186, −6.783090125453255, −6.228626529026482, −5.697063603334090, −5.116436440385964, −4.975125643293245, −4.275293695883246, −3.683039685817496, −3.203981460660511, −2.808023627252977, −2.163398535484330, −1.594454055358186, −0.6718913694363949, 0, 0.6718913694363949, 1.594454055358186, 2.163398535484330, 2.808023627252977, 3.203981460660511, 3.683039685817496, 4.275293695883246, 4.975125643293245, 5.116436440385964, 5.697063603334090, 6.228626529026482, 6.783090125453255, 7.445650051539186, 7.617783898184966, 7.884521920401409, 8.758936258251993, 9.193898288444800, 9.784408587952825, 10.14378482553263, 10.52141759701741, 11.08077437849526, 11.62792061624497, 12.04024433930334, 12.43897490049926, 12.68760937804116

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