Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 149 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3.73·5-s − 2.62·7-s − 8-s + 3.73·10-s + 2.14·11-s − 0.987·13-s + 2.62·14-s + 16-s + 5.24·17-s − 4.24·19-s − 3.73·20-s − 2.14·22-s + 8.14·23-s + 8.94·25-s + 0.987·26-s − 2.62·28-s − 8.43·29-s − 5.21·31-s − 32-s − 5.24·34-s + 9.79·35-s + 0.533·37-s + 4.24·38-s + 3.73·40-s − 3.88·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.66·5-s − 0.991·7-s − 0.353·8-s + 1.18·10-s + 0.646·11-s − 0.273·13-s + 0.701·14-s + 0.250·16-s + 1.27·17-s − 0.973·19-s − 0.834·20-s − 0.456·22-s + 1.69·23-s + 1.78·25-s + 0.193·26-s − 0.495·28-s − 1.56·29-s − 0.937·31-s − 0.176·32-s − 0.899·34-s + 1.65·35-s + 0.0877·37-s + 0.688·38-s + 0.590·40-s − 0.606·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8046} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4721456170$
$L(\frac12)$  $\approx$  $0.4721456170$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;149\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
149 \( 1 - T \)
good5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 + 2.62T + 7T^{2} \)
11 \( 1 - 2.14T + 11T^{2} \)
13 \( 1 + 0.987T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 - 8.14T + 23T^{2} \)
29 \( 1 + 8.43T + 29T^{2} \)
31 \( 1 + 5.21T + 31T^{2} \)
37 \( 1 - 0.533T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 + 3.35T + 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 - 3.22T + 59T^{2} \)
61 \( 1 + 8.02T + 61T^{2} \)
67 \( 1 - 0.811T + 67T^{2} \)
71 \( 1 + 3.95T + 71T^{2} \)
73 \( 1 + 8.88T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + 6.87T + 83T^{2} \)
89 \( 1 - 6.07T + 89T^{2} \)
97 \( 1 + 1.67T + 97T^{2} \)
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

−7.57613048357841381527664889412, −7.41997136184394614614856901560, −6.72398158025621035375049592325, −5.93944458039182522502380481322, −4.99736526235255967635779824002, −4.01873180770407218134581572315, −3.46574356968789223272764593387, −2.88278692101069077496770855811, −1.49174672587043813329586672727, −0.38996306865383215032429799444, 0.38996306865383215032429799444, 1.49174672587043813329586672727, 2.88278692101069077496770855811, 3.46574356968789223272764593387, 4.01873180770407218134581572315, 4.99736526235255967635779824002, 5.93944458039182522502380481322, 6.72398158025621035375049592325, 7.41997136184394614614856901560, 7.57613048357841381527664889412

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