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Properties

Label 2-99-33.26-c2-0-3
Degree $2$
Conductor $99$
Sign $0.961 + 0.273i$
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  + (1.15 − 1.58i)2-s + (0.0432 + 0.133i)4-s + (5.65 + 7.78i)5-s + (−1.61 − 4.97i)7-s + (7.73 + 2.51i)8-s + 18.9·10-s + (−9.06 − 6.23i)11-s + (−13.9 − 10.1i)13-s + (−9.77 − 3.17i)14-s + (12.4 − 9.06i)16-s + (4.53 + 6.24i)17-s + (4.33 − 13.3i)19-s + (−0.792 + 1.09i)20-s + (−20.3 + 7.20i)22-s − 5.68i·23-s + ⋯
L(s)  = 1  + (0.577 − 0.794i)2-s + (0.0108 + 0.0333i)4-s + (1.13 + 1.55i)5-s + (−0.230 − 0.710i)7-s + (0.966 + 0.314i)8-s + 1.89·10-s + (−0.823 − 0.566i)11-s + (−1.07 − 0.778i)13-s + (−0.698 − 0.226i)14-s + (0.779 − 0.566i)16-s + (0.266 + 0.367i)17-s + (0.228 − 0.702i)19-s + (−0.0396 + 0.0545i)20-s + (−0.926 + 0.327i)22-s − 0.247i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.961 + 0.273i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.98883 - 0.277630i\)
\(L(\frac12)\) \(\approx\) \(1.98883 - 0.277630i\)
\(L(2)\) 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 \( 1 + (9.06 + 6.23i)T \)
good2 \( 1 + (-1.15 + 1.58i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-5.65 - 7.78i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (1.61 + 4.97i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (13.9 + 10.1i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-4.53 - 6.24i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-4.33 + 13.3i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 5.68iT - 529T^{2} \)
29 \( 1 + (-22.9 + 7.44i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (12.1 + 8.82i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (2.36 + 7.28i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (17.3 + 5.62i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 53.6T + 1.84e3T^{2} \)
47 \( 1 + (36.9 + 11.9i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (47.9 - 66.0i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-0.000412 + 0.000134i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (3.86 - 2.80i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 111.T + 4.48e3T^{2} \)
71 \( 1 + (-58.5 - 80.6i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (10.5 + 32.6i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-49.9 - 36.2i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-38.3 - 52.7i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 92.5iT - 7.92e3T^{2} \)
97 \( 1 + (-42.4 - 30.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−13.56798570914841473067598832258, −12.76216010194087566203468722174, −11.30409799378663831290818853374, −10.47128982338144363777793308616, −9.965585859823650838548547380195, −7.78377708957071512344432181965, −6.75581232233203980567112733150, −5.26431644930833194694095570707, −3.32294057659503037130117895585, −2.44835558051706330740915482174, 1.88634393488797940384689111199, 4.87462158228102486224044745611, 5.29556251963332115784975831875, 6.54772271546463025440196783984, 8.068862876558008749095415473704, 9.443007984435479230082997932624, 10.06074718606032637217677319323, 12.11053539246453827050827173837, 12.84069625486026098528510125763, 13.75275048194868764108834086045

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