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Properties

Label 2-72450-1.1-c1-0-63
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 − 6·19-s + 2·22-s + 23-s − 2·26-s − 28-s + 8·29-s − 10·31-s − 32-s + 10·37-s + 6·38-s + 10·41-s − 2·43-s − 2·44-s − 46-s − 2·47-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 − 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s + 0.426·22-s + 0.208·23-s − 0.392·26-s − 0.188·28-s + 1.48·29-s − 1.79·31-s − 0.176·32-s + 1.64·37-s + 0.973·38-s + 1.56·41-s − 0.304·43-s − 0.301·44-s − 0.147·46-s − 0.291·47-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.50164334072623, −13.92451052895364, −13.04512744059793, −12.85986878569911, −12.57914875923863, −11.62050348207410, −11.25159368615777, −10.81049558377683, −10.29714304316554, −9.827468664447012, −9.277360364659799, −8.700539857610878, −8.334496389138942, −7.771128954987849, −7.203106306943079, −6.617048235012556, −6.132330822040965, −5.651827118103240, −4.900613794482777, −4.190133309744201, −3.686577031551303, −2.761906722475437, −2.465019605410562, −1.594733930999707, −0.8074089662472190, 0, 0.8074089662472190, 1.594733930999707, 2.465019605410562, 2.761906722475437, 3.686577031551303, 4.190133309744201, 4.900613794482777, 5.651827118103240, 6.132330822040965, 6.617048235012556, 7.203106306943079, 7.771128954987849, 8.334496389138942, 8.700539857610878, 9.277360364659799, 9.827468664447012, 10.29714304316554, 10.81049558377683, 11.25159368615777, 11.62050348207410, 12.57914875923863, 12.85986878569911, 13.04512744059793, 13.92451052895364, 14.50164334072623

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