Properties

Label 2-179520-1.1-c1-0-138
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 $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 3·7-s + 9-s − 11-s + 4·13-s − 15-s + 17-s + 4·19-s − 3·21-s − 4·23-s + 25-s + 27-s − 29-s + 4·31-s − 33-s + 3·35-s + 2·37-s + 4·39-s − 7·41-s − 2·43-s − 45-s − 7·47-s + 2·49-s + 51-s + 11·53-s + 55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 0.718·31-s − 0.174·33-s + 0.507·35-s + 0.328·37-s + 0.640·39-s − 1.09·41-s − 0.304·43-s − 0.149·45-s − 1.02·47-s + 2/7·49-s + 0.140·51-s + 1.51·53-s + 0.134·55-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: \(1\)
Selberg data: \((2,\ 179520,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−13.41666095671890, −13.10721475102908, −12.43762614647533, −12.08610802000953, −11.49476906222466, −11.16918838604368, −10.24993773901976, −10.19701821366717, −9.599835018921550, −9.153445677173192, −8.563720222093690, −8.102898207425849, −7.824247273361277, −7.060778189560073, −6.623430356764522, −6.285937044340694, −5.457927813633709, −5.208710989981123, −4.187895658536573, −3.904182046698732, −3.308463527056713, −2.976184110009126, −2.288582110053830, −1.483768249087926, −0.8223761693481491, 0, 0.8223761693481491, 1.483768249087926, 2.288582110053830, 2.976184110009126, 3.308463527056713, 3.904182046698732, 4.187895658536573, 5.208710989981123, 5.457927813633709, 6.285937044340694, 6.623430356764522, 7.060778189560073, 7.824247273361277, 8.102898207425849, 8.563720222093690, 9.153445677173192, 9.599835018921550, 10.19701821366717, 10.24993773901976, 11.16918838604368, 11.49476906222466, 12.08610802000953, 12.43762614647533, 13.10721475102908, 13.41666095671890

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