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Properties

Label 2-72450-1.1-c1-0-23
Degree $2$
Conductor $72450$
Sign $1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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 − 7-s + 8-s + 2·11-s + 4·13-s − 14-s + 16-s − 2·17-s − 4·19-s + 2·22-s + 23-s + 4·26-s − 28-s + 2·29-s − 4·31-s + 32-s − 2·34-s + 8·37-s − 4·38-s + 2·41-s − 10·43-s + 2·44-s + 46-s − 12·47-s + 49-s + 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.603·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.426·22-s + 0.208·23-s + 0.784·26-s − 0.188·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s + 1.31·37-s − 0.648·38-s + 0.312·41-s − 1.52·43-s + 0.301·44-s + 0.147·46-s − 1.75·47-s + 1/7·49-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.542264644\)
\(L(\frac12)\) \(\approx\) \(3.542264644\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.11604546111294, −13.49612160294348, −13.08721033732153, −12.81382520670468, −12.16398341809601, −11.54956279410156, −11.24569590133436, −10.66526726146837, −10.24917014339622, −9.459672323626322, −9.057035239344372, −8.511272494148351, −7.899146853011158, −7.362661616722846, −6.508915815828458, −6.309213970578309, −6.000166568530474, −4.972217530928293, −4.626569751641099, −3.978579241475447, −3.368087912629522, −2.981595628028523, −1.969368957932323, −1.551165153162127, −0.5388281184878798, 0.5388281184878798, 1.551165153162127, 1.969368957932323, 2.981595628028523, 3.368087912629522, 3.978579241475447, 4.626569751641099, 4.972217530928293, 6.000166568530474, 6.309213970578309, 6.508915815828458, 7.362661616722846, 7.899146853011158, 8.511272494148351, 9.057035239344372, 9.459672323626322, 10.24917014339622, 10.66526726146837, 11.24569590133436, 11.54956279410156, 12.16398341809601, 12.81382520670468, 13.08721033732153, 13.49612160294348, 14.11604546111294

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