Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $0.103 - 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 2.36i)2-s + (0.240 − 1.71i)3-s + (−2.72 + 4.72i)4-s + (0.468 − 0.811i)5-s + (4.38 − 1.77i)6-s + (0.259 + 0.449i)7-s − 9.43·8-s + (−2.88 − 0.824i)9-s + 2.55·10-s + (−0.5 − 0.866i)11-s + (7.44 + 5.81i)12-s + (2.35 − 4.07i)13-s + (−0.708 + 1.22i)14-s + (−1.27 − 0.998i)15-s + (−7.42 − 12.8i)16-s − 2.69·17-s + ⋯
L(s)  = 1  + (0.965 + 1.67i)2-s + (0.138 − 0.990i)3-s + (−1.36 + 2.36i)4-s + (0.209 − 0.362i)5-s + (1.78 − 0.723i)6-s + (0.0981 + 0.169i)7-s − 3.33·8-s + (−0.961 − 0.274i)9-s + 0.808·10-s + (−0.150 − 0.261i)11-s + (2.15 + 1.67i)12-s + (0.652 − 1.13i)13-s + (−0.189 + 0.328i)14-s + (−0.330 − 0.257i)15-s + (−1.85 − 3.21i)16-s − 0.652·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.103 - 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{99} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 99,\ (\ :1/2),\ 0.103 - 0.994i)\)
\(L(1)\)  \(\approx\)  \(1.09565 + 0.987243i\)
\(L(\frac12)\)  \(\approx\)  \(1.09565 + 0.987243i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.240 + 1.71i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.468 + 0.811i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.259 - 0.449i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-2.35 + 4.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 + (3.48 - 6.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.09 + 3.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.59 - 4.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.06T + 37T^{2} \)
41 \( 1 + (0.0865 - 0.149i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.13 - 1.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.153 - 0.266i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 + (1.98 - 3.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.25 + 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.68 - 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.90T + 71T^{2} \)
73 \( 1 - 9.52T + 73T^{2} \)
79 \( 1 + (-1.02 - 1.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.02 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.53T + 89T^{2} \)
97 \( 1 + (8.16 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.00456411771470461055736511416, −13.33466901815405064664020126508, −12.71973119068540871319796106825, −11.55187436075813225073121183834, −9.099314552476444892321316411841, −8.131362132312428003418886563808, −7.32367368439254457301993106412, −6.03338941022192488838967716479, −5.30947180548035457557014545678, −3.38145631767120838025571701785, 2.37980818027998053299919264539, 3.83914380739000482339229521299, 4.74813069034989260249776342890, 6.16206457559506069258825059088, 8.869145879271094285597118586054, 9.781485660471991256072274949371, 10.73319050696359214289445294666, 11.34356608168123863160765210581, 12.46228586938482067854892796466, 13.79564678957594592319639767277

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