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Properties

Label 2-72450-1.1-c1-0-15
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 − 2·13-s − 14-s + 16-s − 4·17-s + 2·19-s − 2·22-s − 23-s − 2·26-s − 28-s + 2·29-s + 10·31-s + 32-s − 4·34-s + 8·37-s + 2·38-s − 6·41-s + 4·43-s − 2·44-s − 46-s − 8·47-s + 49-s − 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s − 0.426·22-s − 0.208·23-s − 0.392·26-s − 0.188·28-s + 0.371·29-s + 1.79·31-s + 0.176·32-s − 0.685·34-s + 1.31·37-s + 0.324·38-s − 0.937·41-s + 0.609·43-s − 0.301·44-s − 0.147·46-s − 1.16·47-s + 1/7·49-s − 0.277·52-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

\(L(1)\) \(\approx\) \(2.744784212\)
\(L(\frac12)\) \(\approx\) \(2.744784212\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.83791841276829, −13.67373436406041, −13.14637494163831, −12.72199047561837, −12.09888695532881, −11.68087142280171, −11.29861887872996, −10.48021336729321, −10.20356864078561, −9.688548221105869, −9.017298014948434, −8.435000661240436, −7.852391255173491, −7.371730004456525, −6.700591083189566, −6.347519244446777, −5.696223818112357, −5.142853753236552, −4.461167979290003, −4.236587879786020, −3.250569313613155, −2.759438646210395, −2.326049944012016, −1.407068784438881, −0.4779401947554412, 0.4779401947554412, 1.407068784438881, 2.326049944012016, 2.759438646210395, 3.250569313613155, 4.236587879786020, 4.461167979290003, 5.142853753236552, 5.696223818112357, 6.347519244446777, 6.700591083189566, 7.371730004456525, 7.852391255173491, 8.435000661240436, 9.017298014948434, 9.688548221105869, 10.20356864078561, 10.48021336729321, 11.29861887872996, 11.68087142280171, 12.09888695532881, 12.72199047561837, 13.14637494163831, 13.67373436406041, 13.83791841276829

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