Properties

Label 2-72450-1.1-c1-0-107
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 − 4·11-s − 2·13-s + 14-s + 16-s + 6·17-s − 4·22-s + 23-s − 2·26-s + 28-s + 2·29-s + 4·31-s + 32-s + 6·34-s − 6·37-s + 6·41-s − 12·43-s − 4·44-s + 46-s − 12·47-s + 49-s − 2·52-s + 6·53-s + 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.852·22-s + 0.208·23-s − 0.392·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.986·37-s + 0.937·41-s − 1.82·43-s − 0.603·44-s + 0.147·46-s − 1.75·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.133·56-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.38543520541491, −13.82417880029099, −13.39972587985485, −12.84262693527397, −12.41454282527449, −11.90007700892863, −11.53597958688538, −10.82134829659624, −10.37070204619277, −9.950142488586063, −9.477512060423580, −8.521618567567872, −8.106517622565210, −7.734632527480819, −7.079149947765393, −6.614710433538812, −5.823086324590405, −5.358612083285335, −4.934422381335114, −4.476815516391392, −3.551074196384802, −3.119976262179713, −2.527235969028953, −1.804955417925684, −1.046876136723577, 0, 1.046876136723577, 1.804955417925684, 2.527235969028953, 3.119976262179713, 3.551074196384802, 4.476815516391392, 4.934422381335114, 5.358612083285335, 5.823086324590405, 6.614710433538812, 7.079149947765393, 7.734632527480819, 8.106517622565210, 8.521618567567872, 9.477512060423580, 9.950142488586063, 10.37070204619277, 10.82134829659624, 11.53597958688538, 11.90007700892863, 12.41454282527449, 12.84262693527397, 13.39972587985485, 13.82417880029099, 14.38543520541491

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