Properties

Label 2-72450-1.1-c1-0-59
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 − 2·11-s − 2·13-s + 14-s + 16-s + 2·17-s − 2·19-s + 2·22-s + 23-s + 2·26-s − 28-s + 6·29-s − 32-s − 2·34-s − 4·37-s + 2·38-s + 10·41-s + 10·43-s − 2·44-s − 46-s − 8·47-s + 49-s − 2·52-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s + 0.426·22-s + 0.208·23-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.176·32-s − 0.342·34-s − 0.657·37-s + 0.324·38-s + 1.56·41-s + 1.52·43-s − 0.301·44-s − 0.147·46-s − 1.16·47-s + 1/7·49-s − 0.277·52-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: $\chi_{72450} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.37698052613813, −14.01541819430954, −13.12395572015937, −12.90045517559641, −12.23200702529123, −11.96228351172858, −11.16172005087425, −10.79071031271093, −10.18972238740016, −9.925486858562670, −9.233881242333672, −8.854530683125282, −8.195841936997558, −7.736309827242702, −7.236786236860484, −6.728142629109511, −6.019154565189199, −5.670504180149880, −4.840218960556444, −4.366747219179964, −3.525501296500137, −2.816919833452738, −2.476695096388468, −1.586983305416349, −0.7969974514984980, 0, 0.7969974514984980, 1.586983305416349, 2.476695096388468, 2.816919833452738, 3.525501296500137, 4.366747219179964, 4.840218960556444, 5.670504180149880, 6.019154565189199, 6.728142629109511, 7.236786236860484, 7.736309827242702, 8.195841936997558, 8.854530683125282, 9.233881242333672, 9.925486858562670, 10.18972238740016, 10.79071031271093, 11.16172005087425, 11.96228351172858, 12.23200702529123, 12.90045517559641, 13.12395572015937, 14.01541819430954, 14.37698052613813

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