Properties

Label 2-72450-1.1-c1-0-10
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 $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 2·11-s + 2·13-s + 14-s + 16-s − 2·19-s + 2·22-s − 23-s − 2·26-s − 28-s − 2·31-s − 32-s + 2·37-s + 2·38-s + 6·41-s − 2·43-s − 2·44-s + 46-s − 2·47-s + 49-s + 2·52-s − 10·53-s + 56-s − 4·59-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 − 0.458·19-s + 0.426·22-s − 0.208·23-s − 0.392·26-s − 0.188·28-s − 0.359·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s + 0.937·41-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 − 1.37·53-s + 0.133·56-s − 0.520·59-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: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.076270385\)
\(L(\frac12)\) \(\approx\) \(1.076270385\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.19611294884095, −13.51832901658453, −13.07987135309664, −12.47947440675749, −12.26162679982573, −11.30796652854491, −11.05771776428784, −10.64619199526155, −9.964529152086187, −9.523433575057022, −9.153297859127929, −8.352986746433559, −8.084007386535552, −7.581183728724969, −6.773525583035435, −6.502703693395563, −5.837883854438697, −5.290738654260324, −4.607070892140168, −3.826819922853189, −3.339665165310832, −2.552909496434677, −2.054098623212342, −1.197774342001081, −0.4090808929404745, 0.4090808929404745, 1.197774342001081, 2.054098623212342, 2.552909496434677, 3.339665165310832, 3.826819922853189, 4.607070892140168, 5.290738654260324, 5.837883854438697, 6.502703693395563, 6.773525583035435, 7.581183728724969, 8.084007386535552, 8.352986746433559, 9.153297859127929, 9.523433575057022, 9.964529152086187, 10.64619199526155, 11.05771776428784, 11.30796652854491, 12.26162679982573, 12.47947440675749, 13.07987135309664, 13.51832901658453, 14.19611294884095

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