Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 1.20·5-s − 4.29·7-s − 8-s + 1.20·10-s + 1.67·11-s − 2.77·13-s + 4.29·14-s + 16-s − 5.75·17-s + 5.66·19-s − 1.20·20-s − 1.67·22-s − 3.89·23-s − 3.55·25-s + 2.77·26-s − 4.29·28-s − 3.66·29-s − 10.2·31-s − 32-s + 5.75·34-s + 5.15·35-s − 7.55·37-s − 5.66·38-s + 1.20·40-s − 1.38·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.537·5-s − 1.62·7-s − 0.353·8-s + 0.379·10-s + 0.504·11-s − 0.768·13-s + 1.14·14-s + 0.250·16-s − 1.39·17-s + 1.30·19-s − 0.268·20-s − 0.356·22-s − 0.812·23-s − 0.711·25-s + 0.543·26-s − 0.811·28-s − 0.680·29-s − 1.83·31-s − 0.176·32-s + 0.986·34-s + 0.871·35-s − 1.24·37-s − 0.919·38-s + 0.189·40-s − 0.216·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1843142877$
$L(\frac12)$  $\approx$  $0.1843142877$
$L(\frac{3}{2})$   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 \( 1 + 1.20T + 5T^{2} \)
7 \( 1 + 4.29T + 7T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 + 2.77T + 13T^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 - 5.66T + 19T^{2} \)
23 \( 1 + 3.89T + 23T^{2} \)
29 \( 1 + 3.66T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 + 1.38T + 41T^{2} \)
43 \( 1 + 7.29T + 43T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 + 0.209T + 53T^{2} \)
59 \( 1 - 1.94T + 59T^{2} \)
61 \( 1 - 1.99T + 61T^{2} \)
67 \( 1 + 2.36T + 67T^{2} \)
71 \( 1 - 9.26T + 71T^{2} \)
73 \( 1 - 7.84T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 3.69T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 5.85T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.73505954231666770752476886187, −7.06907821152116257672030465916, −6.77729999025500770991962371950, −5.89434911611627901312431598543, −5.19461732006031027751137426644, −3.91621384674643423730136787829, −3.58410743766789323999377245152, −2.61760790690085964293964627127, −1.73373678064987525836856913447, −0.22431865172392479639120307821, 0.22431865172392479639120307821, 1.73373678064987525836856913447, 2.61760790690085964293964627127, 3.58410743766789323999377245152, 3.91621384674643423730136787829, 5.19461732006031027751137426644, 5.89434911611627901312431598543, 6.77729999025500770991962371950, 7.06907821152116257672030465916, 7.73505954231666770752476886187

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