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 − 3.80·5-s + 0.515·7-s + 8-s − 3.80·10-s − 0.940·11-s − 2.83·13-s + 0.515·14-s + 16-s + 3.93·17-s − 3.57·19-s − 3.80·20-s − 0.940·22-s + 1.39·23-s + 9.49·25-s − 2.83·26-s + 0.515·28-s + 8.36·29-s + 2.17·31-s + 32-s + 3.93·34-s − 1.96·35-s + 3.67·37-s − 3.57·38-s − 3.80·40-s + 9.56·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.70·5-s + 0.194·7-s + 0.353·8-s − 1.20·10-s − 0.283·11-s − 0.786·13-s + 0.137·14-s + 0.250·16-s + 0.954·17-s − 0.819·19-s − 0.851·20-s − 0.200·22-s + 0.290·23-s + 1.89·25-s − 0.556·26-s + 0.0974·28-s + 1.55·29-s + 0.390·31-s + 0.176·32-s + 0.674·34-s − 0.331·35-s + 0.603·37-s − 0.579·38-s − 0.602·40-s + 1.49·41-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 \( 1 + 3.80T + 5T^{2} \)
7 \( 1 - 0.515T + 7T^{2} \)
11 \( 1 + 0.940T + 11T^{2} \)
13 \( 1 + 2.83T + 13T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + 3.57T + 19T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 - 8.36T + 29T^{2} \)
31 \( 1 - 2.17T + 31T^{2} \)
37 \( 1 - 3.67T + 37T^{2} \)
41 \( 1 - 9.56T + 41T^{2} \)
43 \( 1 + 9.00T + 43T^{2} \)
47 \( 1 + 2.89T + 47T^{2} \)
53 \( 1 + 1.17T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 2.12T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 2.35T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 5.14T + 79T^{2} \)
83 \( 1 - 3.57T + 83T^{2} \)
89 \( 1 - 0.00223T + 89T^{2} \)
97 \( 1 - 3.82T + 97T^{2} \)
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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

−7.61674822588444203684077424863, −6.79345665425127049224314198843, −6.17232191859429650941651056552, −5.04859287621971765608638950550, −4.65782819692063898996004385691, −4.00433725008679726009817933946, −3.17022276823231056170974240188, −2.61441870260798722420396290280, −1.21275825187232792931119436913, 0, 1.21275825187232792931119436913, 2.61441870260798722420396290280, 3.17022276823231056170974240188, 4.00433725008679726009817933946, 4.65782819692063898996004385691, 5.04859287621971765608638950550, 6.17232191859429650941651056552, 6.79345665425127049224314198843, 7.61674822588444203684077424863

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