Properties

Label 2-72450-1.1-c1-0-5
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 − 4·11-s − 3·13-s + 14-s + 16-s + 17-s + 4·22-s + 23-s + 3·26-s − 28-s − 4·31-s − 32-s − 34-s + 11·37-s + 10·41-s − 2·43-s − 4·44-s − 46-s − 11·47-s + 49-s − 3·52-s − 53-s + 56-s + 8·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.852·22-s + 0.208·23-s + 0.588·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1.80·37-s + 1.56·41-s − 0.304·43-s − 0.603·44-s − 0.147·46-s − 1.60·47-s + 1/7·49-s − 0.416·52-s − 0.137·53-s + 0.133·56-s + 1.04·59-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

\(L(1)\) \(\approx\) \(0.6781469860\)
\(L(\frac12)\) \(\approx\) \(0.6781469860\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.28103272872260, −13.34440516276360, −13.13326002734500, −12.60596970982153, −12.10892135788932, −11.50217792013025, −10.93184341602323, −10.61802057943176, −9.934631854241442, −9.541572114189493, −9.246787773676413, −8.338304507942398, −8.004137843851223, −7.499390427276532, −7.043855863299145, −6.375049733582407, −5.785414389617559, −5.272513393934164, −4.644511592852209, −3.963722924803932, −3.071103867813711, −2.689026196406530, −2.091523378987585, −1.187044666863930, −0.3210623265369088, 0.3210623265369088, 1.187044666863930, 2.091523378987585, 2.689026196406530, 3.071103867813711, 3.963722924803932, 4.644511592852209, 5.272513393934164, 5.785414389617559, 6.375049733582407, 7.043855863299145, 7.499390427276532, 8.004137843851223, 8.338304507942398, 9.246787773676413, 9.541572114189493, 9.934631854241442, 10.61802057943176, 10.93184341602323, 11.50217792013025, 12.10892135788932, 12.60596970982153, 13.13326002734500, 13.34440516276360, 14.28103272872260

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