Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s − 4·11-s − 2·13-s + 2·14-s − 4·16-s + 3·17-s − 8·19-s − 2·20-s + 8·22-s − 6·23-s − 4·25-s + 4·26-s − 2·28-s − 4·29-s + 6·31-s + 8·32-s − 6·34-s + 35-s − 3·37-s + 16·38-s + 41-s + 11·43-s − 8·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s − 1.20·11-s − 0.554·13-s + 0.534·14-s − 16-s + 0.727·17-s − 1.83·19-s − 0.447·20-s + 1.70·22-s − 1.25·23-s − 4/5·25-s + 0.784·26-s − 0.377·28-s − 0.742·29-s + 1.07·31-s + 1.41·32-s − 1.02·34-s + 0.169·35-s − 0.493·37-s + 2.59·38-s + 0.156·41-s + 1.67·43-s − 1.20·44-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  =  1
Selberg data  =  $(2,\ 189,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−19.65450874881667, −19.12150399110878, −18.48591100761773, −17.50899608146475, −16.86081391608970, −15.91910494683445, −15.31679160905186, −13.97677627271544, −12.85569129133871, −11.89714523984240, −10.60299964121422, −10.17243898255102, −9.039863529492896, −8.011472352484350, −7.439399344661600, −5.977868951920847, −4.284813946618018, −2.341305950014436, 0, 2.341305950014436, 4.284813946618018, 5.977868951920847, 7.439399344661600, 8.011472352484350, 9.039863529492896, 10.17243898255102, 10.60299964121422, 11.89714523984240, 12.85569129133871, 13.97677627271544, 15.31679160905186, 15.91910494683445, 16.86081391608970, 17.50899608146475, 18.48591100761773, 19.12150399110878, 19.65450874881667

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