Properties

Label 2-265200-1.1-c1-0-54
Degree $2$
Conductor $265200$
Sign $1$
Analytic cond. $2117.63$
Root an. cond. $46.0177$
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  + 3-s + 9-s − 4·11-s − 13-s − 17-s + 8·19-s − 8·23-s + 27-s + 6·29-s − 4·33-s + 6·37-s − 39-s + 6·41-s + 4·43-s + 4·47-s − 7·49-s − 51-s + 10·53-s + 8·57-s + 6·61-s − 8·69-s + 2·73-s − 8·79-s + 81-s + 6·87-s + 2·89-s − 14·97-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 1.83·19-s − 1.66·23-s + 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.986·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.583·47-s − 49-s − 0.140·51-s + 1.37·53-s + 1.05·57-s + 0.768·61-s − 0.963·69-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.643·87-s + 0.211·89-s − 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(2117.63\)
Root analytic conductor: \(46.0177\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{265200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 265200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.180227070\)
\(L(\frac12)\) \(\approx\) \(3.180227070\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.75983686362499, −12.41909961204591, −11.91543547733886, −11.44250055120314, −10.97400855708328, −10.30575284383476, −10.00564546582062, −9.639491448259533, −9.165796212220405, −8.414348913132965, −8.207563766882720, −7.525363185346064, −7.460695710784841, −6.755906601927161, −6.061115683427341, −5.627862384934727, −5.193365379829219, −4.493684758793673, −4.156502808916784, −3.428076674683355, −2.874210874370603, −2.483180305125148, −1.951813927096329, −1.072383727251052, −0.5053511616188375, 0.5053511616188375, 1.072383727251052, 1.951813927096329, 2.483180305125148, 2.874210874370603, 3.428076674683355, 4.156502808916784, 4.493684758793673, 5.193365379829219, 5.627862384934727, 6.061115683427341, 6.755906601927161, 7.460695710784841, 7.525363185346064, 8.207563766882720, 8.414348913132965, 9.165796212220405, 9.639491448259533, 10.00564546582062, 10.30575284383476, 10.97400855708328, 11.44250055120314, 11.91543547733886, 12.41909961204591, 12.75983686362499

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