Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 131 $
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 − 3-s + 4-s − 2·5-s + 6-s + 2·7-s − 8-s + 9-s + 2·10-s − 3·11-s − 12-s + 3·13-s − 2·14-s + 2·15-s + 16-s − 5·17-s − 18-s + 19-s − 2·20-s − 2·21-s + 3·22-s + 4·23-s + 24-s − 25-s − 3·26-s − 27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.904·11-s − 0.288·12-s + 0.832·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.436·21-s + 0.639·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.588·26-s − 0.192·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(786\)    =    \(2 \cdot 3 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{786} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 786,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;131\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;131\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
131 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−10.10212646936986201287332253082, −8.662452261962971816691461028795, −8.367059456801909353888896766196, −7.31589664936853764163787942878, −6.62339846201472136021011440428, −5.35622838853877625371142570288, −4.51306938620050337716148418007, −3.21001886620242484631764818646, −1.63085640210764186035564524216, 0, 1.63085640210764186035564524216, 3.21001886620242484631764818646, 4.51306938620050337716148418007, 5.35622838853877625371142570288, 6.62339846201472136021011440428, 7.31589664936853764163787942878, 8.367059456801909353888896766196, 8.662452261962971816691461028795, 10.10212646936986201287332253082

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