Properties

Degree 2
Conductor $ 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 + 2·5-s + 4·7-s + 3·8-s − 2·10-s − 11-s − 2·13-s − 4·14-s − 16-s + 2·17-s − 2·20-s + 22-s − 8·23-s − 25-s + 2·26-s − 4·28-s + 6·29-s − 8·31-s − 5·32-s − 2·34-s + 8·35-s + 6·37-s + 6·40-s + 2·41-s + 44-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.06·8-s − 0.632·10-s − 0.301·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s + 1.35·35-s + 0.986·37-s + 0.948·40-s + 0.312·41-s + 0.150·44-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{99} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 99,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7923074038$
$L(\frac12)$  $\approx$  $0.7923074038$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.55432357738624, −18.27821831518370, −17.89972372105336, −17.23402150584118, −16.18292498979460, −14.52907875336825, −14.12737143370046, −12.97885466615636, −11.61114553747331, −10.39134307397474, −9.637990407765945, −8.389268452519735, −7.607317265182478, −5.649135945261628, −4.523958719652587, −1.820019998035373, 1.820019998035373, 4.523958719652587, 5.649135945261628, 7.607317265182478, 8.389268452519735, 9.637990407765945, 10.39134307397474, 11.61114553747331, 12.97885466615636, 14.12737143370046, 14.52907875336825, 16.18292498979460, 17.23402150584118, 17.89972372105336, 18.27821831518370, 19.55432357738624

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