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 − 2.72·5-s − 2.79·7-s + 8-s − 2.72·10-s + 3.26·11-s − 2.91·13-s − 2.79·14-s + 16-s + 0.639·17-s + 5.70·19-s − 2.72·20-s + 3.26·22-s − 1.32·23-s + 2.42·25-s − 2.91·26-s − 2.79·28-s − 3.22·29-s − 4.53·31-s + 32-s + 0.639·34-s + 7.62·35-s + 3.32·37-s + 5.70·38-s − 2.72·40-s + 8.09·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.21·5-s − 1.05·7-s + 0.353·8-s − 0.861·10-s + 0.983·11-s − 0.808·13-s − 0.747·14-s + 0.250·16-s + 0.155·17-s + 1.30·19-s − 0.609·20-s + 0.695·22-s − 0.276·23-s + 0.485·25-s − 0.571·26-s − 0.528·28-s − 0.598·29-s − 0.814·31-s + 0.176·32-s + 0.109·34-s + 1.28·35-s + 0.546·37-s + 0.925·38-s − 0.430·40-s + 1.26·41-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 \( 1 + 2.72T + 5T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 - 3.26T + 11T^{2} \)
13 \( 1 + 2.91T + 13T^{2} \)
17 \( 1 - 0.639T + 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 + 1.32T + 23T^{2} \)
29 \( 1 + 3.22T + 29T^{2} \)
31 \( 1 + 4.53T + 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 - 8.09T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 - 0.723T + 59T^{2} \)
61 \( 1 - 3.58T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 16.6T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 5.26T + 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−7.41760173550336213928831835016, −6.89992158814324075238851961360, −5.99330788279110377454183540753, −5.48158186331209222930639297765, −4.33353419562141764374439222385, −4.00950876931415394750876235267, −3.24472516454757243947694563681, −2.60272799542593071084289332055, −1.21678975731420977251942521132, 0, 1.21678975731420977251942521132, 2.60272799542593071084289332055, 3.24472516454757243947694563681, 4.00950876931415394750876235267, 4.33353419562141764374439222385, 5.48158186331209222930639297765, 5.99330788279110377454183540753, 6.89992158814324075238851961360, 7.41760173550336213928831835016

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