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 + 0.102·5-s + 0.350·7-s + 8-s + 0.102·10-s + 2.02·11-s − 0.0298·13-s + 0.350·14-s + 16-s − 3.57·17-s − 2.36·19-s + 0.102·20-s + 2.02·22-s − 3.59·23-s − 4.98·25-s − 0.0298·26-s + 0.350·28-s − 4.55·29-s + 0.839·31-s + 32-s − 3.57·34-s + 0.0357·35-s − 3.75·37-s − 2.36·38-s + 0.102·40-s + 1.43·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.0456·5-s + 0.132·7-s + 0.353·8-s + 0.0322·10-s + 0.611·11-s − 0.00828·13-s + 0.0936·14-s + 0.250·16-s − 0.867·17-s − 0.542·19-s + 0.0228·20-s + 0.432·22-s − 0.748·23-s − 0.997·25-s − 0.00586·26-s + 0.0662·28-s − 0.846·29-s + 0.150·31-s + 0.176·32-s − 0.613·34-s + 0.00604·35-s − 0.617·37-s − 0.383·38-s + 0.0161·40-s + 0.224·41-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 \( 1 - 0.102T + 5T^{2} \)
7 \( 1 - 0.350T + 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 + 0.0298T + 13T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
23 \( 1 + 3.59T + 23T^{2} \)
29 \( 1 + 4.55T + 29T^{2} \)
31 \( 1 - 0.839T + 31T^{2} \)
37 \( 1 + 3.75T + 37T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 + 5.01T + 43T^{2} \)
47 \( 1 - 6.28T + 47T^{2} \)
53 \( 1 + 9.52T + 53T^{2} \)
59 \( 1 + 9.78T + 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 + 4.06T + 67T^{2} \)
71 \( 1 - 9.49T + 71T^{2} \)
73 \( 1 - 7.96T + 73T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 + 9.62T + 83T^{2} \)
89 \( 1 + 0.871T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.45719412184687954717431718426, −6.50747134593484904142325304853, −6.22552702785383277954884296269, −5.35451749543796650488997770517, −4.57091469481466709530258544850, −3.98593296197757115006702589612, −3.26782205360561070293066218534, −2.19290692189994986296235755555, −1.59372808277006851229008052603, 0, 1.59372808277006851229008052603, 2.19290692189994986296235755555, 3.26782205360561070293066218534, 3.98593296197757115006702589612, 4.57091469481466709530258544850, 5.35451749543796650488997770517, 6.22552702785383277954884296269, 6.50747134593484904142325304853, 7.45719412184687954717431718426

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