Properties

Degree 2
Conductor $ 3^{3} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s + 7-s + 6·11-s − 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·20-s − 6·23-s + 4·25-s − 2·28-s − 6·29-s − 4·31-s + 3·35-s − 7·37-s − 3·41-s − 43-s − 12·44-s + 9·47-s + 49-s + 8·52-s − 6·53-s + 18·55-s + 9·59-s − 10·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s + 0.377·7-s + 1.80·11-s − 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.34·20-s − 1.25·23-s + 4/5·25-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.507·35-s − 1.15·37-s − 0.468·41-s − 0.152·43-s − 1.80·44-s + 1.31·47-s + 1/7·49-s + 1.10·52-s − 0.824·53-s + 2.42·55-s + 1.17·59-s − 1.28·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(189\)    =    \(3^{3} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{189} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 189,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.245344016$
$L(\frac12)$  $\approx$  $1.245344016$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.56616752283685, −18.64459683111530, −17.77635473049963, −17.19143261503055, −16.67611701529954, −14.88718645401558, −14.17468510777853, −13.84142406267775, −12.58433731583082, −11.80712467067777, −10.22213033820361, −9.540146555327996, −8.905165354580513, −7.464777888629351, −6.049024665177695, −5.138326337794199, −3.795660929987354, −1.694140236638568, 1.694140236638568, 3.795660929987354, 5.138326337794199, 6.049024665177695, 7.464777888629351, 8.905165354580513, 9.540146555327996, 10.22213033820361, 11.80712467067777, 12.58433731583082, 13.84142406267775, 14.17468510777853, 14.88718645401558, 16.67611701529954, 17.19143261503055, 17.77635473049963, 18.64459683111530, 19.56616752283685

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