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  − 2·4-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  =  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 \( 1 + p T^{2} \)
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

−19.83562821879375, −19.22016578549801, −18.33778184053531, −17.72163508724896, −16.62384435754649, −15.55780525206208, −15.04306097482594, −13.92509300003345, −12.94238601734355, −12.22651025635571, −11.08059552875652, −10.17139469585793, −8.883071919183321, −7.979246979324667, −7.292982517646972, −5.182887133240489, −4.574291127800592, −3.057012925711727, 0, 3.057012925711727, 4.574291127800592, 5.182887133240489, 7.292982517646972, 7.979246979324667, 8.883071919183321, 10.17139469585793, 11.08059552875652, 12.22651025635571, 12.94238601734355, 13.92509300003345, 15.04306097482594, 15.55780525206208, 16.62384435754649, 17.72163508724896, 18.33778184053531, 19.22016578549801, 19.83562821879375

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