Properties

Label 2-320-1.1-c5-0-35
Degree $2$
Conductor $320$
Sign $-1$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 25·5-s + 118·7-s − 207·9-s + 192·11-s − 1.10e3·13-s + 150·15-s + 762·17-s − 2.74e3·19-s + 708·21-s − 1.56e3·23-s + 625·25-s − 2.70e3·27-s − 5.91e3·29-s + 6.86e3·31-s + 1.15e3·33-s + 2.95e3·35-s + 5.51e3·37-s − 6.63e3·39-s − 378·41-s − 2.43e3·43-s − 5.17e3·45-s − 1.31e4·47-s − 2.88e3·49-s + 4.57e3·51-s + 9.17e3·53-s + 4.80e3·55-s + ⋯
L(s)  = 1  + 0.384·3-s + 0.447·5-s + 0.910·7-s − 0.851·9-s + 0.478·11-s − 1.81·13-s + 0.172·15-s + 0.639·17-s − 1.74·19-s + 0.350·21-s − 0.617·23-s + 1/5·25-s − 0.712·27-s − 1.30·29-s + 1.28·31-s + 0.184·33-s + 0.407·35-s + 0.662·37-s − 0.698·39-s − 0.0351·41-s − 0.200·43-s − 0.380·45-s − 0.866·47-s − 0.171·49-s + 0.246·51-s + 0.448·53-s + 0.213·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-1$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - p^{2} T \)
good3 \( 1 - 2 p T + p^{5} T^{2} \)
7 \( 1 - 118 T + p^{5} T^{2} \)
11 \( 1 - 192 T + p^{5} T^{2} \)
13 \( 1 + 1106 T + p^{5} T^{2} \)
17 \( 1 - 762 T + p^{5} T^{2} \)
19 \( 1 + 2740 T + p^{5} T^{2} \)
23 \( 1 + 1566 T + p^{5} T^{2} \)
29 \( 1 + 5910 T + p^{5} T^{2} \)
31 \( 1 - 6868 T + p^{5} T^{2} \)
37 \( 1 - 5518 T + p^{5} T^{2} \)
41 \( 1 + 378 T + p^{5} T^{2} \)
43 \( 1 + 2434 T + p^{5} T^{2} \)
47 \( 1 + 13122 T + p^{5} T^{2} \)
53 \( 1 - 9174 T + p^{5} T^{2} \)
59 \( 1 + 34980 T + p^{5} T^{2} \)
61 \( 1 - 9838 T + p^{5} T^{2} \)
67 \( 1 - 33722 T + p^{5} T^{2} \)
71 \( 1 + 70212 T + p^{5} T^{2} \)
73 \( 1 - 21986 T + p^{5} T^{2} \)
79 \( 1 + 4520 T + p^{5} T^{2} \)
83 \( 1 + 109074 T + p^{5} T^{2} \)
89 \( 1 - 38490 T + p^{5} T^{2} \)
97 \( 1 + 1918 T + p^{5} T^{2} \)
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

−10.23125929903073831051080120522, −9.392455131073455020654360423781, −8.384597401182474782696892267969, −7.65390002399735103922131781849, −6.37189564685972083604974325532, −5.28629953238165554919857601791, −4.27361010433292080968795762092, −2.70458881164568917262651071365, −1.80187145851087221954498523575, 0, 1.80187145851087221954498523575, 2.70458881164568917262651071365, 4.27361010433292080968795762092, 5.28629953238165554919857601791, 6.37189564685972083604974325532, 7.65390002399735103922131781849, 8.384597401182474782696892267969, 9.392455131073455020654360423781, 10.23125929903073831051080120522

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