Properties

Label 2-72450-1.1-c1-0-25
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 + 13-s + 14-s + 16-s − 6·17-s − 6·19-s + 2·22-s + 23-s + 26-s + 28-s + 5·29-s − 8·31-s + 32-s − 6·34-s − 3·37-s − 6·38-s + 9·41-s − 3·43-s + 2·44-s + 46-s + 9·47-s + 49-s + 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 + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.37·19-s + 0.426·22-s + 0.208·23-s + 0.196·26-s + 0.188·28-s + 0.928·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.493·37-s − 0.973·38-s + 1.40·41-s − 0.457·43-s + 0.301·44-s + 0.147·46-s + 1.31·47-s + 1/7·49-s + 0.138·52-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: $\chi_{72450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.686002602\)
\(L(\frac12)\) \(\approx\) \(3.686002602\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−14.15884056454631, −13.62268153546001, −13.03671898356555, −12.74587008747557, −12.14960947928361, −11.65514102340255, −11.04116114374342, −10.77614132993368, −10.33869220675355, −9.427230181667267, −8.930816205645067, −8.593151658930062, −7.947294533736712, −7.157179839271864, −6.885189292600729, −6.191922537124632, −5.851674044119364, −5.064546199488646, −4.459600724010988, −4.125620452442654, −3.542116993497984, −2.647170200370640, −2.140410711234730, −1.528687428048909, −0.5456870125270795, 0.5456870125270795, 1.528687428048909, 2.140410711234730, 2.647170200370640, 3.542116993497984, 4.125620452442654, 4.459600724010988, 5.064546199488646, 5.851674044119364, 6.191922537124632, 6.885189292600729, 7.157179839271864, 7.947294533736712, 8.593151658930062, 8.930816205645067, 9.427230181667267, 10.33869220675355, 10.77614132993368, 11.04116114374342, 11.65514102340255, 12.14960947928361, 12.74587008747557, 13.03671898356555, 13.62268153546001, 14.15884056454631

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