Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 401 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.172·3-s − 5-s + 3.94·7-s − 2.97·9-s + 2.50·11-s + 5.08·13-s + 0.172·15-s − 5.79·17-s − 1.40·19-s − 0.680·21-s − 4.18·23-s + 25-s + 1.02·27-s + 3.24·29-s − 2.87·31-s − 0.431·33-s − 3.94·35-s − 2.92·37-s − 0.876·39-s − 11.1·41-s − 1.07·43-s + 2.97·45-s + 0.742·47-s + 8.59·49-s + 0.998·51-s − 7.39·53-s − 2.50·55-s + ⋯
L(s)  = 1  − 0.0995·3-s − 0.447·5-s + 1.49·7-s − 0.990·9-s + 0.755·11-s + 1.41·13-s + 0.0444·15-s − 1.40·17-s − 0.321·19-s − 0.148·21-s − 0.872·23-s + 0.200·25-s + 0.198·27-s + 0.603·29-s − 0.515·31-s − 0.0751·33-s − 0.667·35-s − 0.480·37-s − 0.140·39-s − 1.73·41-s − 0.163·43-s + 0.442·45-s + 0.108·47-s + 1.22·49-s + 0.139·51-s − 1.01·53-s − 0.337·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8020} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8020,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
401 \( 1 - T \)
good3 \( 1 + 0.172T + 3T^{2} \)
7 \( 1 - 3.94T + 7T^{2} \)
11 \( 1 - 2.50T + 11T^{2} \)
13 \( 1 - 5.08T + 13T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 + 1.40T + 19T^{2} \)
23 \( 1 + 4.18T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 + 2.87T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + 1.07T + 43T^{2} \)
47 \( 1 - 0.742T + 47T^{2} \)
53 \( 1 + 7.39T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 0.250T + 67T^{2} \)
71 \( 1 - 1.16T + 71T^{2} \)
73 \( 1 - 9.19T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 3.29T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.76133803229693405928079354030, −6.52668355508829717703644540930, −6.32506384303791033185827172605, −5.29477223434642541265898653232, −4.64952973366867017585231766451, −3.98123086628997995419731619888, −3.21888571232863784999616935648, −2.03090558005110247079134137473, −1.38523354952156025633250165150, 0, 1.38523354952156025633250165150, 2.03090558005110247079134137473, 3.21888571232863784999616935648, 3.98123086628997995419731619888, 4.64952973366867017585231766451, 5.29477223434642541265898653232, 6.32506384303791033185827172605, 6.52668355508829717703644540930, 7.76133803229693405928079354030

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