Properties

Label 2-72450-1.1-c1-0-30
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 + 6·13-s + 14-s + 16-s + 6·19-s − 23-s − 6·26-s − 28-s − 4·31-s − 32-s + 10·37-s − 6·38-s + 6·41-s − 4·43-s + 46-s + 49-s + 6·52-s − 10·53-s + 56-s + 4·59-s + 4·62-s + 64-s − 2·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.208·23-s − 1.17·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s + 1.64·37-s − 0.973·38-s + 0.937·41-s − 0.609·43-s + 0.147·46-s + 1/7·49-s + 0.832·52-s − 1.37·53-s + 0.133·56-s + 0.520·59-s + 0.508·62-s + 1/8·64-s − 0.237·71-s − 0.234·73-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: $\chi_{72450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.942036926\)
\(L(\frac12)\) \(\approx\) \(1.942036926\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.15615545641194, −13.58921547539000, −12.96735914522285, −12.77938164582695, −11.89960413107313, −11.48723215515410, −11.07904841758100, −10.63189272351505, −9.914332452746661, −9.548447061164874, −9.085311967250384, −8.509571323650276, −8.013567949698381, −7.493157162696387, −6.975621349068312, −6.223166721974591, −5.968253469211772, −5.364125599973676, −4.521663632076665, −3.836112593895015, −3.279907873318964, −2.768223650627030, −1.839701207923373, −1.201561904254499, −0.5764523486128935, 0.5764523486128935, 1.201561904254499, 1.839701207923373, 2.768223650627030, 3.279907873318964, 3.836112593895015, 4.521663632076665, 5.364125599973676, 5.968253469211772, 6.223166721974591, 6.975621349068312, 7.493157162696387, 8.013567949698381, 8.509571323650276, 9.085311967250384, 9.548447061164874, 9.914332452746661, 10.63189272351505, 11.07904841758100, 11.48723215515410, 11.89960413107313, 12.77938164582695, 12.96735914522285, 13.58921547539000, 14.15615545641194

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