Properties

Degree 2
Conductor $ 3^{2} \cdot 19 $
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 + 2·5-s + 3·8-s − 2·10-s + 6·13-s − 16-s + 6·17-s − 19-s − 2·20-s − 4·23-s − 25-s − 6·26-s − 2·29-s + 8·31-s − 5·32-s − 6·34-s − 10·37-s + 38-s + 6·40-s + 2·41-s − 4·43-s + 4·46-s − 12·47-s − 7·49-s + 50-s − 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s + 1.66·13-s − 1/4·16-s + 1.45·17-s − 0.229·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.371·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s − 1.64·37-s + 0.162·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.589·46-s − 1.75·47-s − 49-s + 0.141·50-s − 0.832·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(171\)    =    \(3^{2} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{171} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 171,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8683613404$
$L(\frac12)$  $\approx$  $0.8683613404$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
19 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.65872215970173, −18.82236498214776, −18.14582184330294, −17.50312200760153, −16.63501685602196, −15.81467119146432, −14.36691328780676, −13.70923384888512, −12.98289657399495, −11.63017134596576, −10.33061644135142, −9.849882478281281, −8.700339636508217, −7.962026255736907, −6.378208146887370, −5.302853590311780, −3.699041691023846, −1.484880504698891, 1.484880504698891, 3.699041691023846, 5.302853590311780, 6.378208146887370, 7.962026255736907, 8.700339636508217, 9.849882478281281, 10.33061644135142, 11.63017134596576, 12.98289657399495, 13.70923384888512, 14.36691328780676, 15.81467119146432, 16.63501685602196, 17.50312200760153, 18.14582184330294, 18.82236498214776, 19.65872215970173

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