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Properties

Label 2-99-33.26-c2-0-5
Degree $2$
Conductor $99$
Sign $0.0624 + 0.998i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0624 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0624 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.0624 + 0.998i$
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: \((2,\ 99,\ (\ :1),\ 0.0624 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.334266 - 0.314020i\)
\(L(\frac12)\) \(\approx\) \(0.334266 - 0.314020i\)
\(L(2)\) 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

−13.16206778228027459038811472226, −12.36729725886682062062969755849, −11.55195355320537400662842472797, −9.650893594879373523352605965905, −8.843781345693396542750711327399, −7.71579817917093777770040728770, −7.08316038392691400451409134058, −5.11497385135914673443396226406, −3.80847867008850495669954822766, −0.40889886413435905303349257756, 2.43867964679581155457395124513, 3.72118223635288987448344065436, 6.08713575998361605317736812419, 7.18350547037907172290253981333, 8.640714965126713078066572886771, 9.804913408678658753094333031032, 10.82428642257640913874310464850, 11.66988303726247017740543434800, 12.17067557661379518820267935023, 14.28303164611548390451776401190

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