Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 149 $
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 + 4-s − 1.63·5-s + 4.18·7-s + 8-s − 1.63·10-s − 0.197·11-s − 5.53·13-s + 4.18·14-s + 16-s + 6.24·17-s − 1.76·19-s − 1.63·20-s − 0.197·22-s − 1.25·23-s − 2.32·25-s − 5.53·26-s + 4.18·28-s − 6.46·29-s − 9.67·31-s + 32-s + 6.24·34-s − 6.83·35-s − 8.61·37-s − 1.76·38-s − 1.63·40-s − 7.00·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.731·5-s + 1.58·7-s + 0.353·8-s − 0.517·10-s − 0.0594·11-s − 1.53·13-s + 1.11·14-s + 0.250·16-s + 1.51·17-s − 0.404·19-s − 0.365·20-s − 0.0420·22-s − 0.262·23-s − 0.465·25-s − 1.08·26-s + 0.790·28-s − 1.20·29-s − 1.73·31-s + 0.176·32-s + 1.07·34-s − 1.15·35-s − 1.41·37-s − 0.286·38-s − 0.258·40-s − 1.09·41-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  =  1
Selberg data  =  $(2,\ 8046,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;149\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
149 \( 1 + T \)
good5 \( 1 + 1.63T + 5T^{2} \)
7 \( 1 - 4.18T + 7T^{2} \)
11 \( 1 + 0.197T + 11T^{2} \)
13 \( 1 + 5.53T + 13T^{2} \)
17 \( 1 - 6.24T + 17T^{2} \)
19 \( 1 + 1.76T + 19T^{2} \)
23 \( 1 + 1.25T + 23T^{2} \)
29 \( 1 + 6.46T + 29T^{2} \)
31 \( 1 + 9.67T + 31T^{2} \)
37 \( 1 + 8.61T + 37T^{2} \)
41 \( 1 + 7.00T + 41T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 5.96T + 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 - 6.34T + 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 - 8.52T + 71T^{2} \)
73 \( 1 + 0.842T + 73T^{2} \)
79 \( 1 + 2.79T + 79T^{2} \)
83 \( 1 - 1.68T + 83T^{2} \)
89 \( 1 - 0.518T + 89T^{2} \)
97 \( 1 - 2.99T + 97T^{2} \)
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

−7.53160707018470867042666365844, −7.03976561449997268162130678534, −5.79339093671433242521351454993, −5.21753580964308832604672570207, −4.79913079590025838739779057975, −3.91613459647157127797373669635, −3.34415385522384172646232664811, −2.15006443792993029971209988832, −1.58352882615974936117732675492, 0, 1.58352882615974936117732675492, 2.15006443792993029971209988832, 3.34415385522384172646232664811, 3.91613459647157127797373669635, 4.79913079590025838739779057975, 5.21753580964308832604672570207, 5.79339093671433242521351454993, 7.03976561449997268162130678534, 7.53160707018470867042666365844

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