Properties

Label 2-72450-1.1-c1-0-36
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 − 6·11-s − 2·13-s + 14-s + 16-s − 6·17-s + 2·19-s + 6·22-s + 23-s + 2·26-s − 28-s + 6·29-s + 8·31-s − 32-s + 6·34-s − 8·37-s − 2·38-s − 6·41-s − 2·43-s − 6·44-s − 46-s + 49-s − 2·52-s − 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 1.27·22-s + 0.208·23-s + 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.31·37-s − 0.324·38-s − 0.937·41-s − 0.304·43-s − 0.904·44-s − 0.147·46-s + 1/7·49-s − 0.277·52-s − 1.64·53-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: \(1\)
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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.30380797036247, −13.74389929469114, −13.38374171495138, −12.77259285385300, −12.43781605006901, −11.68071882293531, −11.37418390573723, −10.58622234085723, −10.29712781781926, −9.964588853084029, −9.278880108780039, −8.708678742822853, −8.228401579636796, −7.834527576734443, −7.154436596230064, −6.671447314147272, −6.266841287344142, −5.347597101245206, −4.976829177216775, −4.450036283030068, −3.425300201107648, −2.851382526204125, −2.418042301424138, −1.731826745921210, −0.6523371552714270, 0, 0.6523371552714270, 1.731826745921210, 2.418042301424138, 2.851382526204125, 3.425300201107648, 4.450036283030068, 4.976829177216775, 5.347597101245206, 6.266841287344142, 6.671447314147272, 7.154436596230064, 7.834527576734443, 8.228401579636796, 8.708678742822853, 9.278880108780039, 9.964588853084029, 10.29712781781926, 10.58622234085723, 11.37418390573723, 11.68071882293531, 12.43781605006901, 12.77259285385300, 13.38374171495138, 13.74389929469114, 14.30380797036247

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