Properties

Degree 2
Conductor $ 3 \cdot 43 $
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 + 3-s − 4-s + 2·5-s + 6-s − 3·8-s + 9-s + 2·10-s − 12-s − 2·13-s + 2·15-s − 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s − 4·23-s − 3·24-s − 25-s − 2·26-s + 27-s − 6·29-s + 2·30-s + 8·31-s + 5·32-s − 6·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.834·23-s − 0.612·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.365·30-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(129\)    =    \(3 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{129} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 129,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.609146459$
$L(\frac12)$  $\approx$  $1.609146459$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;43\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
43 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−18.88239856618499, −17.91027763303212, −17.38899703087291, −15.91677312733485, −14.96465544476689, −14.05903853647707, −13.50115305627523, −12.72324052115467, −11.49684764509894, −9.873976645501825, −9.324362356599778, −8.095096812813824, −6.510369299603794, −5.310035288725802, −4.078645038386920, −2.469855667081817, 2.469855667081817, 4.078645038386920, 5.310035288725802, 6.510369299603794, 8.095096812813824, 9.324362356599778, 9.873976645501825, 11.49684764509894, 12.72324052115467, 13.50115305627523, 14.05903853647707, 14.96465544476689, 15.91677312733485, 17.38899703087291, 17.91027763303212, 18.88239856618499

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