Properties

Degree $2$
Conductor $198$
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 − 4·13-s + 2·14-s + 16-s + 6·17-s − 4·19-s + 22-s − 6·23-s − 5·25-s − 4·26-s + 2·28-s − 6·29-s + 8·31-s + 32-s + 6·34-s − 10·37-s − 4·38-s − 6·41-s + 8·43-s + 44-s − 6·46-s + 6·47-s − 3·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.301·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.213·22-s − 1.25·23-s − 25-s − 0.784·26-s + 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.648·38-s − 0.937·41-s + 1.21·43-s + 0.150·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-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

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{198} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.805397590\)
\(L(\frac12)\) \(\approx\) \(1.805397590\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.49664020312762, −18.92016305991576, −17.54772096673954, −17.05169996883540, −15.95918726239770, −14.95684056721627, −14.36088785403802, −13.57057123320379, −12.19135200873569, −11.93886297918362, −10.60919551346141, −9.723620681626334, −8.215106046446206, −7.375620262006203, −6.013653508618363, −4.965244573708906, −3.776726094997060, −2.061673231950376, 2.061673231950376, 3.776726094997060, 4.965244573708906, 6.013653508618363, 7.375620262006203, 8.215106046446206, 9.723620681626334, 10.60919551346141, 11.93886297918362, 12.19135200873569, 13.57057123320379, 14.36088785403802, 14.95684056721627, 15.95918726239770, 17.05169996883540, 17.54772096673954, 18.92016305991576, 19.49664020312762

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