Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11 $
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-s + 4-s − 2·5-s − 4·7-s − 8-s + 2·10-s + 11-s − 6·13-s + 4·14-s + 16-s − 2·17-s + 4·19-s − 2·20-s − 22-s − 4·23-s − 25-s + 6·26-s − 4·28-s − 6·29-s − 32-s + 2·34-s + 8·35-s + 6·37-s − 4·38-s + 2·40-s + 6·41-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 0.353·8-s + 0.632·10-s + 0.301·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.755·28-s − 1.11·29-s − 0.176·32-s + 0.342·34-s + 1.35·35-s + 0.986·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-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  =  1
Selberg data  =  $(2,\ 198,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.88255873437491, −19.20585146386046, −18.43519248149749, −17.27618257285416, −16.53471835198735, −15.78889126025581, −15.11453118828577, −13.89613930072394, −12.56917502864953, −12.06831548180720, −10.95942742879597, −9.695426551233673, −9.331429543089206, −7.761497704345041, −7.142623721416873, −5.904281052539208, −4.093040773818158, −2.740905162257319, 0, 2.740905162257319, 4.093040773818158, 5.904281052539208, 7.142623721416873, 7.761497704345041, 9.331429543089206, 9.695426551233673, 10.95942742879597, 12.06831548180720, 12.56917502864953, 13.89613930072394, 15.11453118828577, 15.78889126025581, 16.53471835198735, 17.27618257285416, 18.43519248149749, 19.20585146386046, 19.88255873437491

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