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Properties

Label 2-72450-1.1-c1-0-119
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 + 14-s + 16-s + 2·17-s + 2·22-s + 23-s + 28-s − 2·29-s − 6·31-s + 32-s + 2·34-s − 4·37-s − 6·41-s + 4·43-s + 2·44-s + 46-s − 12·47-s + 49-s + 56-s − 2·58-s + 6·59-s − 2·61-s − 6·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.426·22-s + 0.208·23-s + 0.188·28-s − 0.371·29-s − 1.07·31-s + 0.176·32-s + 0.342·34-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.301·44-s + 0.147·46-s − 1.75·47-s + 1/7·49-s + 0.133·56-s − 0.262·58-s + 0.781·59-s − 0.256·61-s − 0.762·62-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.22017097707409, −14.10810341908431, −13.20788491103844, −12.97222165221249, −12.42092103657599, −11.73061448790794, −11.57951349034547, −10.95742097778410, −10.40987202574343, −9.886348607358366, −9.304442528382561, −8.688163660110646, −8.254779028142830, −7.438707683458647, −7.220702546295412, −6.480474160803581, −6.003248311078748, −5.358981514668547, −4.924782391363989, −4.319124949110800, −3.569523497818225, −3.326479858831528, −2.386359879641390, −1.732540846773512, −1.161468132055434, 0, 1.161468132055434, 1.732540846773512, 2.386359879641390, 3.326479858831528, 3.569523497818225, 4.319124949110800, 4.924782391363989, 5.358981514668547, 6.003248311078748, 6.480474160803581, 7.220702546295412, 7.438707683458647, 8.254779028142830, 8.688163660110646, 9.304442528382561, 9.886348607358366, 10.40987202574343, 10.95742097778410, 11.57951349034547, 11.73061448790794, 12.42092103657599, 12.97222165221249, 13.20788491103844, 14.10810341908431, 14.22017097707409

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