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  =  $(2,\ 198,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9743637297$
$L(\frac12)$  $\approx$  $0.9743637297$
$L(\frac{3}{2})$   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

−19.51629481036348, −18.44424265107636, −18.00861684212142, −16.96595090836618, −16.32223922426655, −15.26921418526440, −14.43069675623379, −13.50614948610516, −12.11588862961871, −11.51202033453054, −10.41612451305249, −9.554119877489374, −8.369833222628215, −7.679706213784417, −6.372601751119755, −5.123094017219164, −3.427813152453912, −1.514207422128590, 1.514207422128590, 3.427813152453912, 5.123094017219164, 6.372601751119755, 7.679706213784417, 8.369833222628215, 9.554119877489374, 10.41612451305249, 11.51202033453054, 12.11588862961871, 13.50614948610516, 14.43069675623379, 15.26921418526440, 16.32223922426655, 16.96595090836618, 18.00861684212142, 18.44424265107636, 19.51629481036348

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