Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.65·5-s + 0.0358·7-s + 8-s − 1.65·10-s + 1.10·11-s + 1.20·13-s + 0.0358·14-s + 16-s − 3.50·17-s + 4.17·19-s − 1.65·20-s + 1.10·22-s − 5.77·23-s − 2.26·25-s + 1.20·26-s + 0.0358·28-s + 8.50·29-s − 7.39·31-s + 32-s − 3.50·34-s − 0.0592·35-s − 8.83·37-s + 4.17·38-s − 1.65·40-s − 5.77·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.739·5-s + 0.0135·7-s + 0.353·8-s − 0.522·10-s + 0.334·11-s + 0.333·13-s + 0.00958·14-s + 0.250·16-s − 0.849·17-s + 0.958·19-s − 0.369·20-s + 0.236·22-s − 1.20·23-s − 0.453·25-s + 0.235·26-s + 0.00677·28-s + 1.57·29-s − 1.32·31-s + 0.176·32-s − 0.600·34-s − 0.0100·35-s − 1.45·37-s + 0.677·38-s − 0.261·40-s − 0.901·41-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 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 - 0.0358T + 7T^{2} \)
11 \( 1 - 1.10T + 11T^{2} \)
13 \( 1 - 1.20T + 13T^{2} \)
17 \( 1 + 3.50T + 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 + 5.77T + 23T^{2} \)
29 \( 1 - 8.50T + 29T^{2} \)
31 \( 1 + 7.39T + 31T^{2} \)
37 \( 1 + 8.83T + 37T^{2} \)
41 \( 1 + 5.77T + 41T^{2} \)
43 \( 1 + 7.59T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 - 8.49T + 53T^{2} \)
59 \( 1 - 9.24T + 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 3.64T + 73T^{2} \)
79 \( 1 - 4.74T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 + 6.07T + 89T^{2} \)
97 \( 1 - 0.383T + 97T^{2} \)
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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

−7.26706093183654661787880639511, −6.86716993863895731069895133056, −6.03596851877155995771244559882, −5.34488765836497666120298338356, −4.57856670545814723974807772936, −3.84670020747086215753037570791, −3.39411422463403745523608659363, −2.33872161378786591706111269798, −1.41062944589022578218294908970, 0, 1.41062944589022578218294908970, 2.33872161378786591706111269798, 3.39411422463403745523608659363, 3.84670020747086215753037570791, 4.57856670545814723974807772936, 5.34488765836497666120298338356, 6.03596851877155995771244559882, 6.86716993863895731069895133056, 7.26706093183654661787880639511

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