Properties

Label 2-99-33.5-c2-0-3
Degree $2$
Conductor $99$
Sign $0.999 - 0.0152i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 − 0.417i)2-s + (−1.75 + 1.27i)4-s + (4.85 + 1.57i)5-s + (10.6 − 7.73i)7-s + (−4.90 + 6.74i)8-s + 6.89·10-s + (4.16 + 10.1i)11-s + (−0.825 − 2.53i)13-s + (10.4 − 14.3i)14-s + (−0.797 + 2.45i)16-s + (−16.3 − 5.32i)17-s + (−24.7 − 17.9i)19-s + (−10.5 + 3.42i)20-s + (9.60 + 11.3i)22-s + 15.5i·23-s + ⋯
L(s)  = 1  + (0.642 − 0.208i)2-s + (−0.439 + 0.319i)4-s + (0.970 + 0.315i)5-s + (1.52 − 1.10i)7-s + (−0.613 + 0.843i)8-s + 0.689·10-s + (0.378 + 0.925i)11-s + (−0.0634 − 0.195i)13-s + (0.747 − 1.02i)14-s + (−0.0498 + 0.153i)16-s + (−0.963 − 0.312i)17-s + (−1.30 − 0.947i)19-s + (−0.527 + 0.171i)20-s + (0.436 + 0.515i)22-s + 0.675i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.999 - 0.0152i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.999 - 0.0152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.94528 + 0.0148059i\)
\(L(\frac12)\) \(\approx\) \(1.94528 + 0.0148059i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-4.16 - 10.1i)T \)
good2 \( 1 + (-1.28 + 0.417i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (-4.85 - 1.57i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (-10.6 + 7.73i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (0.825 + 2.53i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (16.3 + 5.32i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (24.7 + 17.9i)T + (111. + 343. i)T^{2} \)
23 \( 1 - 15.5iT - 529T^{2} \)
29 \( 1 + (6.10 + 8.39i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-5.21 - 16.0i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (15.1 - 11.0i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-7.85 + 10.8i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 10.5T + 1.84e3T^{2} \)
47 \( 1 + (48.3 - 66.5i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (-59.9 + 19.4i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (38.7 + 53.3i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-11.2 + 34.7i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 60.5T + 4.48e3T^{2} \)
71 \( 1 + (46.7 + 15.2i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (5.15 - 3.74i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-35.5 - 109. i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-38.5 - 12.5i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 71.1iT - 7.92e3T^{2} \)
97 \( 1 + (-5.76 - 17.7i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.71852229110129226376902569854, −12.89798764147561533181387603953, −11.53568746805861524364127538728, −10.66198521390927370188891310596, −9.354491108825185124364445468536, −8.090036663071268269197205741620, −6.77250411561054791358988263738, −5.03486078393134452333646962323, −4.21992261010248118717039217403, −2.10764036771116463231361358922, 1.91182374682204558844481186731, 4.34626849008460025213405951436, 5.50158540244143939286385861094, 6.21918205240925539375224867648, 8.487022755977473946259321691751, 9.010830621650136506499702483402, 10.48380912183790534836121137651, 11.70998626882388492915754848761, 12.83708861925201068438627684781, 13.78172217294911903681400012768

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