Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
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  + 0.803·2-s − 3-s − 1.35·4-s − 0.904·5-s − 0.803·6-s + 1.61·7-s − 2.69·8-s + 9-s − 0.727·10-s − 1.65·11-s + 1.35·12-s − 0.0916·13-s + 1.29·14-s + 0.904·15-s + 0.539·16-s − 17-s + 0.803·18-s − 5.68·19-s + 1.22·20-s − 1.61·21-s − 1.32·22-s + 5.14·23-s + 2.69·24-s − 4.18·25-s − 0.0736·26-s − 27-s − 2.18·28-s + ⋯
L(s)  = 1  + 0.568·2-s − 0.577·3-s − 0.676·4-s − 0.404·5-s − 0.328·6-s + 0.608·7-s − 0.953·8-s + 0.333·9-s − 0.230·10-s − 0.498·11-s + 0.390·12-s − 0.0254·13-s + 0.346·14-s + 0.233·15-s + 0.134·16-s − 0.242·17-s + 0.189·18-s − 1.30·19-s + 0.273·20-s − 0.351·21-s − 0.283·22-s + 1.07·23-s + 0.550·24-s − 0.836·25-s − 0.0144·26-s − 0.192·27-s − 0.412·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 - 0.803T + 2T^{2} \)
5 \( 1 + 0.904T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 1.65T + 11T^{2} \)
13 \( 1 + 0.0916T + 13T^{2} \)
19 \( 1 + 5.68T + 19T^{2} \)
23 \( 1 - 5.14T + 23T^{2} \)
29 \( 1 - 7.70T + 29T^{2} \)
31 \( 1 - 4.00T + 31T^{2} \)
37 \( 1 - 9.54T + 37T^{2} \)
41 \( 1 - 0.791T + 41T^{2} \)
43 \( 1 + 1.21T + 43T^{2} \)
47 \( 1 + 0.249T + 47T^{2} \)
53 \( 1 + 2.79T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 + 1.36T + 61T^{2} \)
67 \( 1 - 8.35T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 6.03T + 73T^{2} \)
79 \( 1 - 4.64T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 0.693T + 97T^{2} \)
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

−7.52386447869975581718186856827, −6.55130084527476953142914026741, −6.05209744939979817737748433114, −5.22224256051065541552566707004, −4.52375485628677203867027668530, −4.33824269321254699953058164603, −3.23704304808308747906956012274, −2.38548760613051141696104181157, −1.06141174369463085617705111212, 0, 1.06141174369463085617705111212, 2.38548760613051141696104181157, 3.23704304808308747906956012274, 4.33824269321254699953058164603, 4.52375485628677203867027668530, 5.22224256051065541552566707004, 6.05209744939979817737748433114, 6.55130084527476953142914026741, 7.52386447869975581718186856827

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