Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $0.998 - 0.0475i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.190 + 0.587i)2-s + (1.30 − 0.951i)4-s + (0.809 − 2.48i)5-s + (−2.42 + 1.76i)7-s + (1.80 + 1.31i)8-s + 1.61·10-s + (−1.69 + 2.85i)11-s + (0.545 + 1.67i)13-s + (−1.5 − 1.08i)14-s + (0.572 − 1.76i)16-s + (−0.5 + 1.53i)17-s + (−4.73 − 3.44i)19-s + (−1.30 − 4.02i)20-s + (−2 − 0.449i)22-s − 3.47·23-s + ⋯
L(s)  = 1  + (0.135 + 0.415i)2-s + (0.654 − 0.475i)4-s + (0.361 − 1.11i)5-s + (−0.917 + 0.666i)7-s + (0.639 + 0.464i)8-s + 0.511·10-s + (−0.509 + 0.860i)11-s + (0.151 + 0.465i)13-s + (−0.400 − 0.291i)14-s + (0.143 − 0.440i)16-s + (−0.121 + 0.373i)17-s + (−1.08 − 0.789i)19-s + (−0.292 − 0.900i)20-s + (−0.426 − 0.0957i)22-s − 0.723·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.998 - 0.0475i$
motivic weight  =  \(1\)
character  :  $\chi_{99} (82, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 99,\ (\ :1/2),\ 0.998 - 0.0475i)$
$L(1)$  $\approx$  $1.18014 + 0.0280619i$
$L(\frac12)$  $\approx$  $1.18014 + 0.0280619i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (1.69 - 2.85i)T \)
good2 \( 1 + (-0.190 - 0.587i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.809 + 2.48i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.42 - 1.76i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.545 - 1.67i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.5 - 1.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (-3.61 + 2.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.881 - 2.71i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.190 - 0.138i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.66 + 7.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + (-1.30 - 0.951i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.97 - 9.14i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.35 + 6.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.42 + 7.46i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (-1.71 + 5.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.61 + 1.90i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.92 - 9.00i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.218 - 0.673i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.527T + 89T^{2} \)
97 \( 1 + (4.33 + 13.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.88441208634309124846948143435, −12.81953525050876356491018718170, −12.08559870458602472863358782503, −10.58136597216823745583855793868, −9.535510910893011390947133214239, −8.465345796978454265012026473221, −6.87911670244084392931576428502, −5.86792800328731985688302816538, −4.69391520472387922687432997586, −2.17771360057825058654361214910, 2.67353381586304762334914147412, 3.70461679290982177190831233442, 6.13731191355275439606919306151, 6.95837837767768320332728360660, 8.215873627560383150253341548128, 10.21656361328303341069277793850, 10.51361359276888462709231394160, 11.68294092597360301387349458332, 12.94933566608028165844851750419, 13.64976981721785308814852644527

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