Properties

Label 2-179520-1.1-c1-0-25
Degree $2$
Conductor $179520$
Sign $1$
Analytic cond. $1433.47$
Root an. cond. $37.8612$
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  + 3-s + 5-s − 2·7-s + 9-s − 11-s + 15-s + 17-s − 2·21-s − 4·23-s + 25-s + 27-s − 2·29-s + 4·31-s − 33-s − 2·35-s + 2·37-s + 6·43-s + 45-s − 3·49-s + 51-s + 6·53-s − 55-s − 14·59-s + 2·61-s − 2·63-s − 14·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.242·17-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s − 0.338·35-s + 0.328·37-s + 0.914·43-s + 0.149·45-s − 3/7·49-s + 0.140·51-s + 0.824·53-s − 0.134·55-s − 1.82·59-s + 0.256·61-s − 0.251·63-s − 1.71·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(179520\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(1433.47\)
Root analytic conductor: \(37.8612\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{179520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 179520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.437263847\)
\(L(\frac12)\) \(\approx\) \(2.437263847\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−13.32017679085073, −12.65749719453333, −12.33729048736249, −11.95605596856742, −11.12254791382922, −10.79077807789568, −10.18142088496252, −9.783341084469215, −9.442951394056983, −8.960547475808588, −8.365892275343526, −7.906914924627569, −7.441800387544292, −6.888258787289763, −6.265755166831558, −5.977240860442821, −5.359748181487782, −4.706368184723167, −4.160800373459549, −3.590538662574585, −3.001490491371568, −2.560756824901060, −1.943509416901192, −1.264792745001273, −0.4294892504538646, 0.4294892504538646, 1.264792745001273, 1.943509416901192, 2.560756824901060, 3.001490491371568, 3.590538662574585, 4.160800373459549, 4.706368184723167, 5.359748181487782, 5.977240860442821, 6.265755166831558, 6.888258787289763, 7.441800387544292, 7.906914924627569, 8.365892275343526, 8.960547475808588, 9.442951394056983, 9.783341084469215, 10.18142088496252, 10.79077807789568, 11.12254791382922, 11.95605596856742, 12.33729048736249, 12.65749719453333, 13.32017679085073

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