Properties

Degree 2
Conductor 6047
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  − 2.65·2-s − 2.99·3-s + 5.04·4-s − 2.08·5-s + 7.95·6-s − 1.11·7-s − 8.08·8-s + 5.97·9-s + 5.52·10-s + 3.38·11-s − 15.1·12-s − 3.23·13-s + 2.96·14-s + 6.23·15-s + 11.3·16-s + 4.29·17-s − 15.8·18-s + 0.715·19-s − 10.4·20-s + 3.34·21-s − 8.98·22-s + 0.981·23-s + 24.2·24-s − 0.671·25-s + 8.59·26-s − 8.89·27-s − 5.62·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 1.72·3-s + 2.52·4-s − 0.930·5-s + 3.24·6-s − 0.421·7-s − 2.85·8-s + 1.99·9-s + 1.74·10-s + 1.02·11-s − 4.36·12-s − 0.897·13-s + 0.791·14-s + 1.60·15-s + 2.84·16-s + 1.04·17-s − 3.73·18-s + 0.164·19-s − 2.34·20-s + 0.729·21-s − 1.91·22-s + 0.204·23-s + 4.94·24-s − 0.134·25-s + 1.68·26-s − 1.71·27-s − 1.06·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6047\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6047} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6047,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 6047$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 6047$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad6047 \( 1+O(T) \)
good2 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 + 2.99T + 3T^{2} \)
5 \( 1 + 2.08T + 5T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 - 3.38T + 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 - 4.29T + 17T^{2} \)
19 \( 1 - 0.715T + 19T^{2} \)
23 \( 1 - 0.981T + 23T^{2} \)
29 \( 1 + 2.22T + 29T^{2} \)
31 \( 1 - 0.355T + 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + 1.13T + 41T^{2} \)
43 \( 1 + 6.13T + 43T^{2} \)
47 \( 1 + 2.54T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 7.65T + 61T^{2} \)
67 \( 1 + 9.28T + 67T^{2} \)
71 \( 1 + 3.24T + 71T^{2} \)
73 \( 1 + 4.55T + 73T^{2} \)
79 \( 1 + 2.45T + 79T^{2} \)
83 \( 1 - 7.49T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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

−7.68129206606107186409089084960, −7.12105577743389990076655208163, −6.52506671036979906413293193550, −5.97694276984673170817422158572, −5.05881280248955248549508885154, −4.04722785558020424213668100231, −3.03488618240444469014331907768, −1.64592786974979312212520740141, −0.803470463471443516605543451320, 0, 0.803470463471443516605543451320, 1.64592786974979312212520740141, 3.03488618240444469014331907768, 4.04722785558020424213668100231, 5.05881280248955248549508885154, 5.97694276984673170817422158572, 6.52506671036979906413293193550, 7.12105577743389990076655208163, 7.68129206606107186409089084960

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