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  − 2.32·3-s − 5-s + 0.515·7-s + 2.38·9-s + 4.54·11-s + 1.42·13-s + 2.32·15-s + 3.21·17-s − 2.56·19-s − 1.19·21-s − 0.925·23-s + 25-s + 1.41·27-s − 1.44·29-s − 6.16·31-s − 10.5·33-s − 0.515·35-s − 5.88·37-s − 3.29·39-s − 9.02·41-s − 7.54·43-s − 2.38·45-s − 1.06·47-s − 6.73·49-s − 7.46·51-s + 3.39·53-s − 4.54·55-s + ⋯
L(s)  = 1  − 1.34·3-s − 0.447·5-s + 0.194·7-s + 0.796·9-s + 1.36·11-s + 0.394·13-s + 0.599·15-s + 0.779·17-s − 0.589·19-s − 0.261·21-s − 0.192·23-s + 0.200·25-s + 0.272·27-s − 0.268·29-s − 1.10·31-s − 1.83·33-s − 0.0871·35-s − 0.966·37-s − 0.528·39-s − 1.40·41-s − 1.15·43-s − 0.356·45-s − 0.156·47-s − 0.961·49-s − 1.04·51-s + 0.466·53-s − 0.612·55-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 \( 1 + 2.32T + 3T^{2} \)
7 \( 1 - 0.515T + 7T^{2} \)
11 \( 1 - 4.54T + 11T^{2} \)
13 \( 1 - 1.42T + 13T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
19 \( 1 + 2.56T + 19T^{2} \)
23 \( 1 + 0.925T + 23T^{2} \)
29 \( 1 + 1.44T + 29T^{2} \)
31 \( 1 + 6.16T + 31T^{2} \)
37 \( 1 + 5.88T + 37T^{2} \)
41 \( 1 + 9.02T + 41T^{2} \)
43 \( 1 + 7.54T + 43T^{2} \)
47 \( 1 + 1.06T + 47T^{2} \)
53 \( 1 - 3.39T + 53T^{2} \)
59 \( 1 - 8.53T + 59T^{2} \)
61 \( 1 - 4.29T + 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 6.48T + 73T^{2} \)
79 \( 1 - 6.18T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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

−7.27182227478616663748261108834, −6.60779662100161509432794708672, −6.24208981359304027120963945942, −5.30068427253479100530005306911, −4.91610045119244416998237359464, −3.83287299330737000233476214654, −3.49328264546655901632436313389, −1.91837407689772630585300363021, −1.09247427952807017573011427261, 0, 1.09247427952807017573011427261, 1.91837407689772630585300363021, 3.49328264546655901632436313389, 3.83287299330737000233476214654, 4.91610045119244416998237359464, 5.30068427253479100530005306911, 6.24208981359304027120963945942, 6.60779662100161509432794708672, 7.27182227478616663748261108834

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