Properties

Degree 2
Conductor $ 2 \cdot 3^{4} $
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 + 2·7-s − 8-s + 3·11-s + 2·13-s − 2·14-s + 16-s + 3·17-s − 19-s − 3·22-s + 6·23-s − 5·25-s − 2·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 3·34-s − 4·37-s + 38-s − 9·41-s − 43-s + 3·44-s − 6·46-s + 6·47-s − 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.639·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s + 0.162·38-s − 1.40·41-s − 0.152·43-s + 0.452·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(162\)    =    \(2 \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{162} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 162,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9335592475$
$L(\frac12)$  $\approx$  $0.9335592475$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.00688277120738, −18.42746631330654, −17.26128934152204, −16.93118765065978, −15.70555831584594, −14.82782448622343, −13.98195129207708, −12.70434390882728, −11.57619177441088, −10.96157598355786, −9.711604924732819, −8.789487803962223, −7.806708544758987, −6.686197134032186, −5.343839769637045, −3.650760604853421, −1.606932511063427, 1.606932511063427, 3.650760604853421, 5.343839769637045, 6.686197134032186, 7.806708544758987, 8.789487803962223, 9.711604924732819, 10.96157598355786, 11.57619177441088, 12.70434390882728, 13.98195129207708, 14.82782448622343, 15.70555831584594, 16.93118765065978, 17.26128934152204, 18.42746631330654, 19.00688277120738

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