Properties

Label 2-72450-1.1-c1-0-80
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 + 4·13-s + 14-s + 16-s + 2·19-s + 23-s − 4·26-s − 28-s − 6·29-s + 2·31-s − 32-s + 10·37-s − 2·38-s − 6·41-s + 4·43-s − 46-s + 6·47-s + 49-s + 4·52-s − 6·53-s + 56-s + 6·58-s − 12·59-s − 10·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.458·19-s + 0.208·23-s − 0.784·26-s − 0.188·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.64·37-s − 0.324·38-s − 0.937·41-s + 0.609·43-s − 0.147·46-s + 0.875·47-s + 1/7·49-s + 0.554·52-s − 0.824·53-s + 0.133·56-s + 0.787·58-s − 1.56·59-s − 1.28·61-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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
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 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.32241212685152, −13.86125057160088, −13.24418153221366, −12.95859135606073, −12.24020583104742, −11.81041964839854, −11.13711581414148, −10.91492616491640, −10.31209909852144, −9.719913884330231, −9.215213953043378, −8.928203167429432, −8.220298934059558, −7.700207529097513, −7.318985763997712, −6.570596168457611, −6.054948866103270, −5.759638100346882, −4.870372730083250, −4.237837030505001, −3.503580679978660, −3.060318000560835, −2.306888066796735, −1.507849813677678, −0.9277381608301906, 0, 0.9277381608301906, 1.507849813677678, 2.306888066796735, 3.060318000560835, 3.503580679978660, 4.237837030505001, 4.870372730083250, 5.759638100346882, 6.054948866103270, 6.570596168457611, 7.318985763997712, 7.700207529097513, 8.220298934059558, 8.928203167429432, 9.215213953043378, 9.719913884330231, 10.31209909852144, 10.91492616491640, 11.13711581414148, 11.81041964839854, 12.24020583104742, 12.95859135606073, 13.24418153221366, 13.86125057160088, 14.32241212685152

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