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 + 1.53·5-s + 5.04·7-s − 8-s − 1.53·10-s + 2.19·11-s − 3.71·13-s − 5.04·14-s + 16-s + 2.18·17-s + 0.428·19-s + 1.53·20-s − 2.19·22-s + 2.68·23-s − 2.65·25-s + 3.71·26-s + 5.04·28-s + 1.03·29-s + 0.812·31-s − 32-s − 2.18·34-s + 7.71·35-s − 2.71·37-s − 0.428·38-s − 1.53·40-s + 3.18·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.684·5-s + 1.90·7-s − 0.353·8-s − 0.483·10-s + 0.662·11-s − 1.03·13-s − 1.34·14-s + 0.250·16-s + 0.530·17-s + 0.0982·19-s + 0.342·20-s − 0.468·22-s + 0.560·23-s − 0.531·25-s + 0.728·26-s + 0.953·28-s + 0.191·29-s + 0.145·31-s − 0.176·32-s − 0.375·34-s + 1.30·35-s − 0.446·37-s − 0.0694·38-s − 0.241·40-s + 0.497·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.337385106$
$L(\frac12)$  $\approx$  $2.337385106$
$L(\frac{3}{2})$   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 \( 1 - 1.53T + 5T^{2} \)
7 \( 1 - 5.04T + 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + 3.71T + 13T^{2} \)
17 \( 1 - 2.18T + 17T^{2} \)
19 \( 1 - 0.428T + 19T^{2} \)
23 \( 1 - 2.68T + 23T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 - 0.812T + 31T^{2} \)
37 \( 1 + 2.71T + 37T^{2} \)
41 \( 1 - 3.18T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 - 3.13T + 47T^{2} \)
53 \( 1 + 7.68T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 0.845T + 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 - 9.74T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 + 1.50T + 83T^{2} \)
89 \( 1 - 6.94T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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

−7.80760415973534824039164433465, −7.37551499443240194878160834324, −6.58483246545734310636203110011, −5.70457767618963797460567977708, −5.07410209450069013922215027610, −4.50050774497275336901118520412, −3.38277592662236407306341051415, −2.23671850238021672710213780553, −1.76282584463544001255483445161, −0.889519521614863996493417070378, 0.889519521614863996493417070378, 1.76282584463544001255483445161, 2.23671850238021672710213780553, 3.38277592662236407306341051415, 4.50050774497275336901118520412, 5.07410209450069013922215027610, 5.70457767618963797460567977708, 6.58483246545734310636203110011, 7.37551499443240194878160834324, 7.80760415973534824039164433465

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