Properties

Degree 2
Conductor $ 19 \cdot 317 $
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  − 2.79·2-s − 1.66·3-s + 5.78·4-s + 1.47·5-s + 4.65·6-s − 1.60·7-s − 10.5·8-s − 0.220·9-s − 4.11·10-s + 1.76·11-s − 9.65·12-s + 3.00·13-s + 4.48·14-s − 2.45·15-s + 17.9·16-s − 0.637·17-s + 0.616·18-s − 19-s + 8.53·20-s + 2.67·21-s − 4.91·22-s − 6.91·23-s + 17.6·24-s − 2.82·25-s − 8.37·26-s + 5.36·27-s − 9.29·28-s + ⋯
L(s)  = 1  − 1.97·2-s − 0.962·3-s + 2.89·4-s + 0.659·5-s + 1.89·6-s − 0.606·7-s − 3.73·8-s − 0.0736·9-s − 1.30·10-s + 0.531·11-s − 2.78·12-s + 0.832·13-s + 1.19·14-s − 0.634·15-s + 4.48·16-s − 0.154·17-s + 0.145·18-s − 0.229·19-s + 1.90·20-s + 0.584·21-s − 1.04·22-s − 1.44·23-s + 3.59·24-s − 0.565·25-s − 1.64·26-s + 1.03·27-s − 1.75·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6023\)    =    \(19 \cdot 317\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6023} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6023,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;317\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;317\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 + T \)
317 \( 1 + T \)
good2 \( 1 + 2.79T + 2T^{2} \)
3 \( 1 + 1.66T + 3T^{2} \)
5 \( 1 - 1.47T + 5T^{2} \)
7 \( 1 + 1.60T + 7T^{2} \)
11 \( 1 - 1.76T + 11T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
17 \( 1 + 0.637T + 17T^{2} \)
23 \( 1 + 6.91T + 23T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 - 4.71T + 31T^{2} \)
37 \( 1 + 7.76T + 37T^{2} \)
41 \( 1 - 2.91T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 8.67T + 53T^{2} \)
59 \( 1 + 9.03T + 59T^{2} \)
61 \( 1 + 0.869T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 8.83T + 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 4.35T + 89T^{2} \)
97 \( 1 + 7.99T + 97T^{2} \)
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

−7.987664566241754380124110265482, −6.88506157202854880774547617089, −6.35165609003885759202566946646, −6.15364301418477439535642975838, −5.30548813740412952260332125286, −3.80168736553976223874582664626, −2.79153795796899366748743226755, −1.89736237453136980781783325112, −0.994298653082831483749030155516, 0, 0.994298653082831483749030155516, 1.89736237453136980781783325112, 2.79153795796899366748743226755, 3.80168736553976223874582664626, 5.30548813740412952260332125286, 6.15364301418477439535642975838, 6.35165609003885759202566946646, 6.88506157202854880774547617089, 7.987664566241754380124110265482

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