Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11 $
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 − 11-s + 2·13-s + 2·14-s + 16-s − 6·17-s + 2·19-s − 22-s − 5·25-s + 2·26-s + 2·28-s − 6·29-s − 4·31-s + 32-s − 6·34-s + 2·37-s + 2·38-s + 6·41-s − 10·43-s − 44-s + 12·47-s − 3·49-s − 5·50-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.301·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.213·22-s − 25-s + 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s − 1.52·43-s − 0.150·44-s + 1.75·47-s − 3/7·49-s − 0.707·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{198} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 198,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.804931829$
$L(\frac12)$  $\approx$  $1.804931829$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.81506761579472, −18.57807787723614, −17.85024252745043, −16.91960851836072, −15.80955522761737, −15.23979916970512, −14.20875646828420, −13.45942332771753, −12.62322523514586, −11.35225328119077, −11.00040472824991, −9.548038803864552, −8.329295392613665, −7.297312500161406, −6.049775125269170, −4.947125709928331, −3.782169051293808, −2.060782848680638, 2.060782848680638, 3.782169051293808, 4.947125709928331, 6.049775125269170, 7.297312500161406, 8.329295392613665, 9.548038803864552, 11.00040472824991, 11.35225328119077, 12.62322523514586, 13.45942332771753, 14.20875646828420, 15.23979916970512, 15.80955522761737, 16.91960851836072, 17.85024252745043, 18.57807787723614, 19.81506761579472

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