Properties

Label 2-320-1.1-c1-0-7
Degree $2$
Conductor $320$
Sign $-1$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s − 6·37-s − 6·41-s + 8·43-s + 3·45-s + 4·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s + 2·61-s + 12·63-s − 2·65-s − 8·67-s − 6·73-s + 16·77-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s − 0.977·67-s − 0.702·73-s + 1.82·77-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-1$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.01108015186625578670651780817, −10.39655063289957855259191225662, −9.206637608661514184739644068270, −8.421702762202697929017375581976, −7.28963827504896405934712065825, −6.22671236180701133336548488807, −5.28966562607622137666656232279, −3.64413717916439174649981742998, −2.75667504176180688667939525456, 0, 2.75667504176180688667939525456, 3.64413717916439174649981742998, 5.28966562607622137666656232279, 6.22671236180701133336548488807, 7.28963827504896405934712065825, 8.421702762202697929017375581976, 9.206637608661514184739644068270, 10.39655063289957855259191225662, 11.01108015186625578670651780817

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