Properties

Degree 2
Conductor $ 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-s − 2.24·3-s + 4-s − 5-s + 2.24·6-s − 2.96·7-s − 8-s + 2.03·9-s + 10-s + 3.78·11-s − 2.24·12-s + 1.59·13-s + 2.96·14-s + 2.24·15-s + 16-s − 1.08·17-s − 2.03·18-s − 1.19·19-s − 20-s + 6.65·21-s − 3.78·22-s − 9.20·23-s + 2.24·24-s + 25-s − 1.59·26-s + 2.16·27-s − 2.96·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.29·3-s + 0.5·4-s − 0.447·5-s + 0.916·6-s − 1.12·7-s − 0.353·8-s + 0.678·9-s + 0.316·10-s + 1.14·11-s − 0.647·12-s + 0.441·13-s + 0.793·14-s + 0.579·15-s + 0.250·16-s − 0.263·17-s − 0.479·18-s − 0.275·19-s − 0.223·20-s + 1.45·21-s − 0.806·22-s − 1.91·23-s + 0.458·24-s + 0.200·25-s − 0.312·26-s + 0.416·27-s − 0.560·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4010\)    =    \(2 \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 + 2.24T + 3T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
13 \( 1 - 1.59T + 13T^{2} \)
17 \( 1 + 1.08T + 17T^{2} \)
19 \( 1 + 1.19T + 19T^{2} \)
23 \( 1 + 9.20T + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 7.57T + 31T^{2} \)
37 \( 1 - 3.06T + 37T^{2} \)
41 \( 1 + 0.377T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 - 3.63T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 - 9.43T + 67T^{2} \)
71 \( 1 + 6.13T + 71T^{2} \)
73 \( 1 + 0.754T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 3.77T + 83T^{2} \)
89 \( 1 - 5.67T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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

−8.165639531424325988034285547134, −7.12417354740629773622923277189, −6.52931502007893308951480643127, −6.14414418622588569895748267744, −5.35831402127468791397287574897, −4.11662359793209497165230728875, −3.60379587935780582933898709563, −2.25100378865759688174797999899, −0.943273183984849138103240408056, 0, 0.943273183984849138103240408056, 2.25100378865759688174797999899, 3.60379587935780582933898709563, 4.11662359793209497165230728875, 5.35831402127468791397287574897, 6.14414418622588569895748267744, 6.52931502007893308951480643127, 7.12417354740629773622923277189, 8.165639531424325988034285547134

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