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  + 2.57·3-s − 5-s + 0.619·7-s + 3.61·9-s − 4.29·11-s + 3.48·13-s − 2.57·15-s − 3.08·17-s − 5.05·19-s + 1.59·21-s − 4.10·23-s + 25-s + 1.59·27-s + 6.51·29-s − 0.284·31-s − 11.0·33-s − 0.619·35-s + 4.51·37-s + 8.96·39-s − 6.30·41-s − 3.21·43-s − 3.61·45-s − 1.32·47-s − 6.61·49-s − 7.93·51-s − 0.392·53-s + 4.29·55-s + ⋯
L(s)  = 1  + 1.48·3-s − 0.447·5-s + 0.234·7-s + 1.20·9-s − 1.29·11-s + 0.966·13-s − 0.664·15-s − 0.748·17-s − 1.15·19-s + 0.347·21-s − 0.856·23-s + 0.200·25-s + 0.306·27-s + 1.21·29-s − 0.0510·31-s − 1.92·33-s − 0.104·35-s + 0.741·37-s + 1.43·39-s − 0.984·41-s − 0.490·43-s − 0.539·45-s − 0.193·47-s − 0.945·49-s − 1.11·51-s − 0.0538·53-s + 0.579·55-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 \( 1 - 2.57T + 3T^{2} \)
7 \( 1 - 0.619T + 7T^{2} \)
11 \( 1 + 4.29T + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 3.08T + 17T^{2} \)
19 \( 1 + 5.05T + 19T^{2} \)
23 \( 1 + 4.10T + 23T^{2} \)
29 \( 1 - 6.51T + 29T^{2} \)
31 \( 1 + 0.284T + 31T^{2} \)
37 \( 1 - 4.51T + 37T^{2} \)
41 \( 1 + 6.30T + 41T^{2} \)
43 \( 1 + 3.21T + 43T^{2} \)
47 \( 1 + 1.32T + 47T^{2} \)
53 \( 1 + 0.392T + 53T^{2} \)
59 \( 1 + 8.91T + 59T^{2} \)
61 \( 1 - 4.78T + 61T^{2} \)
67 \( 1 - 4.97T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + 9.58T + 73T^{2} \)
79 \( 1 + 1.38T + 79T^{2} \)
83 \( 1 - 1.04T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 5.28T + 97T^{2} \)
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

−7.910179411184899035003128098887, −6.88706572406435159523604344347, −6.30282718339369180377420311286, −5.25360398478524342446542275589, −4.40528318445789938281739393151, −3.85807539562239856187607546192, −2.98625811984031000289419584470, −2.41134768629659306944145298514, −1.56503940487372944419387662519, 0, 1.56503940487372944419387662519, 2.41134768629659306944145298514, 2.98625811984031000289419584470, 3.85807539562239856187607546192, 4.40528318445789938281739393151, 5.25360398478524342446542275589, 6.30282718339369180377420311286, 6.88706572406435159523604344347, 7.910179411184899035003128098887

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