↑

Properties

Label 2-221760-1.1-c1-0-253
Degree $2$
Conductor $221760$
Sign $-1$
Analytic cond. $1770.76$
Root an. cond. $42.0804$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 11-s − 2·13-s + 6·17-s − 8·19-s + 6·23-s + 25-s + 6·29-s + 2·31-s − 35-s − 2·37-s − 8·43-s + 12·47-s + 49-s + 6·53-s + 55-s + 6·59-s − 8·61-s + 2·65-s − 2·67-s − 10·73-s − 77-s + 8·79-s + 12·83-s − 6·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.169·35-s − 0.328·37-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.781·59-s − 1.02·61-s + 0.248·65-s − 0.244·67-s − 1.17·73-s − 0.113·77-s + 0.900·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221760\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1770.76\)
Root analytic conductor: \(42.0804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.23691076071639, −12.53927190556675, −12.28055132243656, −11.94321943925512, −11.34417902469849, −10.78722250625636, −10.39223495962096, −10.13510746609496, −9.449028183626291, −8.748884868260670, −8.571773718899393, −8.020494434293893, −7.467367541814010, −7.140938844365194, −6.504133058325835, −6.041491150354040, −5.349997278859664, −4.922598170933131, −4.498733675261991, −3.872887161446562, −3.324139907338271, −2.665345714423335, −2.266128387529589, −1.374864450215793, −0.8229396055077944, 0, 0.8229396055077944, 1.374864450215793, 2.266128387529589, 2.665345714423335, 3.324139907338271, 3.872887161446562, 4.498733675261991, 4.922598170933131, 5.349997278859664, 6.041491150354040, 6.504133058325835, 7.140938844365194, 7.467367541814010, 8.020494434293893, 8.571773718899393, 8.748884868260670, 9.449028183626291, 10.13510746609496, 10.39223495962096, 10.78722250625636, 11.34417902469849, 11.94321943925512, 12.28055132243656, 12.53927190556675, 13.23691076071639

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