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  + 2·2-s + 2·4-s − 5-s − 2·7-s − 2·10-s − 11-s + 4·13-s − 4·14-s − 4·16-s + 2·17-s − 2·20-s − 2·22-s + 23-s − 4·25-s + 8·26-s − 4·28-s + 7·31-s − 8·32-s + 4·34-s + 2·35-s + 3·37-s + 8·41-s − 6·43-s − 2·44-s + 2·46-s − 8·47-s − 3·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s − 0.632·10-s − 0.301·11-s + 1.10·13-s − 1.06·14-s − 16-s + 0.485·17-s − 0.447·20-s − 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.755·28-s + 1.25·31-s − 1.41·32-s + 0.685·34-s + 0.338·35-s + 0.493·37-s + 1.24·41-s − 0.914·43-s − 0.301·44-s + 0.294·46-s − 1.16·47-s − 3/7·49-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  =  $(2,\ 99,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.684496332$
$L(\frac12)$  $\approx$  $1.684496332$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.84772284531684, −18.89838904942361, −17.84691079076185, −16.32014828245270, −15.71727940010753, −14.81456437027325, −13.68040274047176, −13.05532878430428, −12.07025537765624, −11.14136051051457, −9.701226498296212, −8.228854358433494, −6.665039873048448, −5.664076600247276, −4.190851151743579, −3.090009165928871, 3.090009165928871, 4.190851151743579, 5.664076600247276, 6.665039873048448, 8.228854358433494, 9.701226498296212, 11.14136051051457, 12.07025537765624, 13.05532878430428, 13.68040274047176, 14.81456437027325, 15.71727940010753, 16.32014828245270, 17.84691079076185, 18.89838904942361, 19.84772284531684

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