↑

Properties

Label 2-72450-1.1-c1-0-14
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

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 − 4·19-s + 2·22-s + 23-s − 2·26-s − 28-s − 2·29-s − 8·31-s + 32-s − 2·34-s − 4·37-s − 4·38-s + 12·41-s − 2·43-s + 2·44-s + 46-s + 8·47-s + 49-s − 2·52-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 − 0.917·19-s + 0.426·22-s + 0.208·23-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.657·37-s − 0.648·38-s + 1.87·41-s − 0.304·43-s + 0.301·44-s + 0.147·46-s + 1.16·47-s + 1/7·49-s − 0.277·52-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: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.723204471\)
\(L(\frac12)\) \(\approx\) \(2.723204471\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.19302275213832, −13.50995690277906, −13.09328784878314, −12.66689424139089, −12.21118268073275, −11.69523691913417, −11.16477826142958, −10.58619447631349, −10.30567914569161, −9.413250523919763, −9.085983056542348, −8.619563018894192, −7.752908874898950, −7.274525578472950, −6.854160393737797, −6.250468717822856, −5.703235759951298, −5.256223714269218, −4.359714917666759, −4.132058829116717, −3.479088793974683, −2.712720451233660, −2.193116237583948, −1.494250245076854, −0.4659139194537798, 0.4659139194537798, 1.494250245076854, 2.193116237583948, 2.712720451233660, 3.479088793974683, 4.132058829116717, 4.359714917666759, 5.256223714269218, 5.703235759951298, 6.250468717822856, 6.854160393737797, 7.274525578472950, 7.752908874898950, 8.619563018894192, 9.085983056542348, 9.413250523919763, 10.30567914569161, 10.58619447631349, 11.16477826142958, 11.69523691913417, 12.21118268073275, 12.66689424139089, 13.09328784878314, 13.50995690277906, 14.19302275213832

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