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  =  $(2,\ 162,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.683969137$
$L(\frac12)$  $\approx$  $1.683969137$
$L(\frac{3}{2})$   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

−19.37592613666697, −18.10310895796481, −17.64497963807767, −16.21276886235263, −15.65962998061819, −14.63893489064142, −13.78279575403141, −12.99772192968520, −11.89683285365063, −11.04396588795957, −10.11405722912124, −8.543319215757721, −7.636714974661113, −6.259365937774684, −5.138053261109394, −3.948473300195872, −2.198793856118012, 2.198793856118012, 3.948473300195872, 5.138053261109394, 6.259365937774684, 7.636714974661113, 8.543319215757721, 10.11405722912124, 11.04396588795957, 11.89683285365063, 12.99772192968520, 13.78279575403141, 14.63893489064142, 15.65962998061819, 16.21276886235263, 17.64497963807767, 18.10310895796481, 19.37592613666697

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