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 − 0.683·5-s − 0.639·7-s + 8-s − 0.683·10-s − 1.37·11-s + 1.22·13-s − 0.639·14-s + 16-s − 2.93·17-s − 1.25·19-s − 0.683·20-s − 1.37·22-s + 7.55·23-s − 4.53·25-s + 1.22·26-s − 0.639·28-s − 2.51·29-s + 5.78·31-s + 32-s − 2.93·34-s + 0.436·35-s − 2.36·37-s − 1.25·38-s − 0.683·40-s − 1.79·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.305·5-s − 0.241·7-s + 0.353·8-s − 0.216·10-s − 0.415·11-s + 0.340·13-s − 0.170·14-s + 0.250·16-s − 0.712·17-s − 0.288·19-s − 0.152·20-s − 0.293·22-s + 1.57·23-s − 0.906·25-s + 0.240·26-s − 0.120·28-s − 0.466·29-s + 1.03·31-s + 0.176·32-s − 0.503·34-s + 0.0738·35-s − 0.389·37-s − 0.203·38-s − 0.108·40-s − 0.279·41-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 \( 1 + 0.683T + 5T^{2} \)
7 \( 1 + 0.639T + 7T^{2} \)
11 \( 1 + 1.37T + 11T^{2} \)
13 \( 1 - 1.22T + 13T^{2} \)
17 \( 1 + 2.93T + 17T^{2} \)
19 \( 1 + 1.25T + 19T^{2} \)
23 \( 1 - 7.55T + 23T^{2} \)
29 \( 1 + 2.51T + 29T^{2} \)
31 \( 1 - 5.78T + 31T^{2} \)
37 \( 1 + 2.36T + 37T^{2} \)
41 \( 1 + 1.79T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 3.25T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 1.02T + 61T^{2} \)
67 \( 1 - 8.00T + 67T^{2} \)
71 \( 1 - 2.68T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 - 2.29T + 83T^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 - 0.495T + 97T^{2} \)
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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

−7.28418803949060901657172348088, −6.72737885958308634395614094201, −6.14972464759606050283047754122, −5.21298207053719401269488587850, −4.75844618633161486138974432185, −3.85944455099924691703773738802, −3.22689379766815340820490517518, −2.41960644768587308284598281818, −1.41582879122772396150512583547, 0, 1.41582879122772396150512583547, 2.41960644768587308284598281818, 3.22689379766815340820490517518, 3.85944455099924691703773738802, 4.75844618633161486138974432185, 5.21298207053719401269488587850, 6.14972464759606050283047754122, 6.72737885958308634395614094201, 7.28418803949060901657172348088

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