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Properties

Label 2-72450-1.1-c1-0-96
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 $1$

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 − 2·17-s − 4·19-s + 2·22-s + 23-s − 2·26-s − 28-s + 8·29-s + 2·31-s + 32-s − 2·34-s − 4·37-s − 4·38-s + 2·41-s + 8·43-s + 2·44-s + 46-s − 12·47-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 − 0.485·17-s − 0.917·19-s + 0.426·22-s + 0.208·23-s − 0.392·26-s − 0.188·28-s + 1.48·29-s + 0.359·31-s + 0.176·32-s − 0.342·34-s − 0.657·37-s − 0.648·38-s + 0.312·41-s + 1.21·43-s + 0.301·44-s + 0.147·46-s − 1.75·47-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: \(1\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ -1)\)

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

−14.37702643799255, −13.81804925271734, −13.45966878326361, −12.68559834613092, −12.46380589444239, −12.08370003439373, −11.27514648595724, −11.03952738675709, −10.36072963377388, −9.889351842793234, −9.292611808229834, −8.808138081469276, −8.120068717484142, −7.695989749522725, −6.823237773415145, −6.569393779705315, −6.173461967482046, −5.397479299957755, −4.665440364499580, −4.487634744956176, −3.674673594631840, −3.109972657520786, −2.482369830914346, −1.853847452717773, −1.000654308460642, 0, 1.000654308460642, 1.853847452717773, 2.482369830914346, 3.109972657520786, 3.674673594631840, 4.487634744956176, 4.665440364499580, 5.397479299957755, 6.173461967482046, 6.569393779705315, 6.823237773415145, 7.695989749522725, 8.120068717484142, 8.808138081469276, 9.292611808229834, 9.889351842793234, 10.36072963377388, 11.03952738675709, 11.27514648595724, 12.08370003439373, 12.46380589444239, 12.68559834613092, 13.45966878326361, 13.81804925271734, 14.37702643799255

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