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 + 4·5-s − 2·7-s − 3·8-s + 4·10-s + 11-s − 2·13-s − 2·14-s − 16-s − 2·17-s − 6·19-s − 4·20-s + 22-s − 4·23-s + 11·25-s − 2·26-s + 2·28-s + 6·29-s + 4·31-s + 5·32-s − 2·34-s − 8·35-s − 6·37-s − 6·38-s − 12·40-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.78·5-s − 0.755·7-s − 1.06·8-s + 1.26·10-s + 0.301·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.894·20-s + 0.213·22-s − 0.834·23-s + 11/5·25-s − 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s − 1.35·35-s − 0.986·37-s − 0.973·38-s − 1.89·40-s + 1.56·41-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  =  $(2,\ 99,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.364292295$
$L(\frac12)$  $\approx$  $1.364292295$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
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

−19.66740203000470, −18.62895480472995, −17.53905693607894, −17.20600292443617, −15.73958136507882, −14.42614897696167, −13.90490267946748, −12.98872560550180, −12.34498791478187, −10.45031744709830, −9.602153864880357, −8.765403563089384, −6.524958699189111, −5.826428735085359, −4.446766565285289, −2.561416597493789, 2.561416597493789, 4.446766565285289, 5.826428735085359, 6.524958699189111, 8.765403563089384, 9.602153864880357, 10.45031744709830, 12.34498791478187, 12.98872560550180, 13.90490267946748, 14.42614897696167, 15.73958136507882, 17.20600292443617, 17.53905693607894, 18.62895480472995, 19.66740203000470

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