Properties

Label 2-8020-1.1-c1-0-127
Degree $2$
Conductor $8020$
Sign $-1$
Analytic cond. $64.0400$
Root an. cond. $8.00250$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.73·3-s + 5-s − 4.22·7-s + 4.50·9-s + 4.03·11-s − 0.502·13-s + 2.73·15-s − 3.20·17-s − 0.594·19-s − 11.5·21-s − 6.87·23-s + 25-s + 4.12·27-s − 5.10·29-s − 5.67·31-s + 11.0·33-s − 4.22·35-s − 5.99·37-s − 1.37·39-s − 9.59·41-s − 1.01·43-s + 4.50·45-s − 3.60·47-s + 10.8·49-s − 8.77·51-s − 0.850·53-s + 4.03·55-s + ⋯
L(s)  = 1  + 1.58·3-s + 0.447·5-s − 1.59·7-s + 1.50·9-s + 1.21·11-s − 0.139·13-s + 0.707·15-s − 0.776·17-s − 0.136·19-s − 2.52·21-s − 1.43·23-s + 0.200·25-s + 0.793·27-s − 0.948·29-s − 1.01·31-s + 1.92·33-s − 0.714·35-s − 0.984·37-s − 0.220·39-s − 1.49·41-s − 0.154·43-s + 0.671·45-s − 0.526·47-s + 1.55·49-s − 1.22·51-s − 0.116·53-s + 0.544·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
Sign: $-1$
Analytic conductor: \(64.0400\)
Root analytic conductor: \(8.00250\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8020,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
401 \( 1 + T \)
good3 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 + 4.22T + 7T^{2} \)
11 \( 1 - 4.03T + 11T^{2} \)
13 \( 1 + 0.502T + 13T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 + 0.594T + 19T^{2} \)
23 \( 1 + 6.87T + 23T^{2} \)
29 \( 1 + 5.10T + 29T^{2} \)
31 \( 1 + 5.67T + 31T^{2} \)
37 \( 1 + 5.99T + 37T^{2} \)
41 \( 1 + 9.59T + 41T^{2} \)
43 \( 1 + 1.01T + 43T^{2} \)
47 \( 1 + 3.60T + 47T^{2} \)
53 \( 1 + 0.850T + 53T^{2} \)
59 \( 1 - 9.79T + 59T^{2} \)
61 \( 1 + 4.10T + 61T^{2} \)
67 \( 1 - 1.02T + 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 9.35T + 73T^{2} \)
79 \( 1 + 7.27T + 79T^{2} \)
83 \( 1 - 0.745T + 83T^{2} \)
89 \( 1 + 7.78T + 89T^{2} \)
97 \( 1 - 7.26T + 97T^{2} \)
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

−7.44297963867536205846794812049, −6.81860793380728458434597736187, −6.35805257112239308556068167259, −5.51453512509709860251318986243, −4.27197510423779193299124444726, −3.63276882244979640835391526763, −3.25956657156116902555988011210, −2.22406422907631873923430879626, −1.70243556558266213111594028781, 0, 1.70243556558266213111594028781, 2.22406422907631873923430879626, 3.25956657156116902555988011210, 3.63276882244979640835391526763, 4.27197510423779193299124444726, 5.51453512509709860251318986243, 6.35805257112239308556068167259, 6.81860793380728458434597736187, 7.44297963867536205846794812049

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