Properties

Degree 2
Conductor $ 3^{2} \cdot 11 $
Sign $0.545 + 0.838i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 1.86i)2-s + (−0.635 − 1.61i)3-s + (−1.32 − 2.29i)4-s + (−1.81 − 3.13i)5-s + (3.69 + 0.550i)6-s + (1.13 − 1.96i)7-s + 1.39·8-s + (−2.19 + 2.04i)9-s + 7.81·10-s + (−0.5 + 0.866i)11-s + (−2.85 + 3.58i)12-s + (−0.619 − 1.07i)13-s + (2.44 + 4.23i)14-s + (−3.90 + 4.91i)15-s + (1.14 − 1.98i)16-s + 5.69·17-s + ⋯
L(s)  = 1  + (−0.762 + 1.32i)2-s + (−0.366 − 0.930i)3-s + (−0.661 − 1.14i)4-s + (−0.810 − 1.40i)5-s + (1.50 + 0.224i)6-s + (0.429 − 0.743i)7-s + 0.492·8-s + (−0.730 + 0.682i)9-s + 2.47·10-s + (−0.150 + 0.261i)11-s + (−0.823 + 1.03i)12-s + (−0.171 − 0.297i)13-s + (0.654 + 1.13i)14-s + (−1.00 + 1.26i)15-s + (0.286 − 0.495i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.545 + 0.838i$
motivic weight  =  \(1\)
character  :  $\chi_{99} (34, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 99,\ (\ :1/2),\ 0.545 + 0.838i)\)
\(L(1)\)  \(\approx\)  \(0.406643 - 0.220626i\)
\(L(\frac12)\)  \(\approx\)  \(0.406643 - 0.220626i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.635 + 1.61i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.07 - 1.86i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.81 + 3.13i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.13 + 1.96i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.619 + 1.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.69T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + (2.95 + 5.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (-4.98 - 8.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.57 + 6.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.34 + 7.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.16T + 53T^{2} \)
59 \( 1 + (1.45 + 2.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.13 - 5.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.68 + 8.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 4.31T + 73T^{2} \)
79 \( 1 + (0.708 - 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.37 + 2.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.77T + 89T^{2} \)
97 \( 1 + (-1.27 + 2.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.88820550180425615922470065802, −12.58971666681266366980497958140, −11.94849774078633707559539573124, −10.30483925835444743127085595539, −8.702901172203196726415357815155, −7.974017023043989234759629850325, −7.35431802402363051393858774595, −5.89825725737881031304735430165, −4.67025176343136594905340310929, −0.74853076515329875380635694455, 2.77760752557498646334963294429, 3.86843007748427001023293042077, 5.88593171555966075386645531454, 7.76826253068596008981390347522, 9.022349019410599760778603775452, 10.14598930348002935118959637153, 10.84158293943188274932993400117, 11.63775572938113064529535992369, 12.20828000343087903018011817646, 14.33315307249265669838506551771

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