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Properties

Label 2-72450-1.1-c1-0-13
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 − 13-s − 14-s + 16-s + 17-s − 23-s + 26-s + 28-s − 4·29-s + 4·31-s − 32-s − 34-s − 3·37-s − 2·41-s + 6·43-s + 46-s − 47-s + 49-s − 52-s + 5·53-s − 56-s + 4·58-s − 8·61-s − 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.208·23-s + 0.196·26-s + 0.188·28-s − 0.742·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.493·37-s − 0.312·41-s + 0.914·43-s + 0.147·46-s − 0.145·47-s + 1/7·49-s − 0.138·52-s + 0.686·53-s − 0.133·56-s + 0.525·58-s − 1.02·61-s − 0.508·62-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: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.440246029\)
\(L(\frac12)\) \(\approx\) \(1.440246029\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.09651243399630, −13.75317410854749, −12.95367289824156, −12.60743882917558, −11.90046693477697, −11.64030910648951, −11.03316378712696, −10.48034763636831, −10.12812710096534, −9.482328579386153, −9.053998512276151, −8.529674947909752, −7.847885926542496, −7.669225924542455, −6.882959698663343, −6.503569555578339, −5.703441088882683, −5.331590042424999, −4.571239321153428, −3.965691680327544, −3.266675196195793, −2.569379723716528, −1.959085187356899, −1.265694389059468, −0.4657939436208743, 0.4657939436208743, 1.265694389059468, 1.959085187356899, 2.569379723716528, 3.266675196195793, 3.965691680327544, 4.571239321153428, 5.331590042424999, 5.703441088882683, 6.503569555578339, 6.882959698663343, 7.669225924542455, 7.847885926542496, 8.529674947909752, 9.053998512276151, 9.482328579386153, 10.12812710096534, 10.48034763636831, 11.03316378712696, 11.64030910648951, 11.90046693477697, 12.60743882917558, 12.95367289824156, 13.75317410854749, 14.09651243399630

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