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 + 1.99·5-s − 0.0206·7-s − 8-s − 1.99·10-s + 5.26·11-s + 1.30·13-s + 0.0206·14-s + 16-s − 4.05·17-s + 0.295·19-s + 1.99·20-s − 5.26·22-s − 1.64·23-s − 1.02·25-s − 1.30·26-s − 0.0206·28-s − 3.96·29-s + 8.69·31-s − 32-s + 4.05·34-s − 0.0410·35-s − 8.66·37-s − 0.295·38-s − 1.99·40-s − 0.291·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.891·5-s − 0.00779·7-s − 0.353·8-s − 0.630·10-s + 1.58·11-s + 0.362·13-s + 0.00551·14-s + 0.250·16-s − 0.983·17-s + 0.0677·19-s + 0.445·20-s − 1.12·22-s − 0.343·23-s − 0.205·25-s − 0.256·26-s − 0.00389·28-s − 0.736·29-s + 1.56·31-s − 0.176·32-s + 0.695·34-s − 0.00694·35-s − 1.42·37-s − 0.0478·38-s − 0.315·40-s − 0.0455·41-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  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.947747665$
$L(\frac12)$  $\approx$  $1.947747665$
$L(\frac{3}{2})$   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 \( 1 - 1.99T + 5T^{2} \)
7 \( 1 + 0.0206T + 7T^{2} \)
11 \( 1 - 5.26T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 + 4.05T + 17T^{2} \)
19 \( 1 - 0.295T + 19T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 - 8.69T + 31T^{2} \)
37 \( 1 + 8.66T + 37T^{2} \)
41 \( 1 + 0.291T + 41T^{2} \)
43 \( 1 + 0.389T + 43T^{2} \)
47 \( 1 - 4.57T + 47T^{2} \)
53 \( 1 - 7.77T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 - 5.96T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 5.21T + 73T^{2} \)
79 \( 1 - 5.69T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 0.168T + 97T^{2} \)
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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

−7.966455747743893157110711244190, −6.94071798093901683632959943025, −6.58184165672622087750285060588, −5.98812722458299921156510669411, −5.18826217084870599704475604542, −4.15828822321138802700017343531, −3.50971313510353864983203578355, −2.31321622736677897828254887432, −1.75264059172404144745645352301, −0.791413120403158780761639582602, 0.791413120403158780761639582602, 1.75264059172404144745645352301, 2.31321622736677897828254887432, 3.50971313510353864983203578355, 4.15828822321138802700017343531, 5.18826217084870599704475604542, 5.98812722458299921156510669411, 6.58184165672622087750285060588, 6.94071798093901683632959943025, 7.966455747743893157110711244190

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