↑

Properties

Label 2-99-11.8-c4-0-10
Degree $2$
Conductor $99$
Sign $0.678 - 0.734i$
Analytic cond. $10.2336$
Root an. cond. $3.19900$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.46 + 3.39i)2-s + (−0.502 + 1.54i)4-s + (−14.7 − 10.7i)5-s + (47.9 + 15.5i)7-s + (57.3 − 18.6i)8-s − 76.4i·10-s + (82.9 + 88.1i)11-s + (125. + 173. i)13-s + (65.4 + 201. i)14-s + (226. + 164. i)16-s + (255. − 351. i)17-s + (−61.3 + 19.9i)19-s + (23.9 − 17.4i)20-s + (−94.5 + 499. i)22-s − 536.·23-s + ⋯
L(s)  = 1  + (0.616 + 0.849i)2-s + (−0.0314 + 0.0966i)4-s + (−0.589 − 0.428i)5-s + (0.979 + 0.318i)7-s + (0.896 − 0.291i)8-s − 0.764i·10-s + (0.685 + 0.728i)11-s + (0.744 + 1.02i)13-s + (0.334 + 1.02i)14-s + (0.882 + 0.641i)16-s + (0.884 − 1.21i)17-s + (−0.170 + 0.0552i)19-s + (0.0599 − 0.0435i)20-s + (−0.195 + 1.03i)22-s − 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(10.2336\)
Root analytic conductor: \(3.19900\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :2),\ 0.678 - 0.734i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.44821 + 1.07100i\)
\(L(\frac12)\) \(\approx\) \(2.44821 + 1.07100i\)
\(L(3)\) 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 \( 1 + (-82.9 - 88.1i)T \)
good2 \( 1 + (-2.46 - 3.39i)T + (-4.94 + 15.2i)T^{2} \)
5 \( 1 + (14.7 + 10.7i)T + (193. + 594. i)T^{2} \)
7 \( 1 + (-47.9 - 15.5i)T + (1.94e3 + 1.41e3i)T^{2} \)
13 \( 1 + (-125. - 173. i)T + (-8.82e3 + 2.71e4i)T^{2} \)
17 \( 1 + (-255. + 351. i)T + (-2.58e4 - 7.94e4i)T^{2} \)
19 \( 1 + (61.3 - 19.9i)T + (1.05e5 - 7.66e4i)T^{2} \)
23 \( 1 + 536.T + 2.79e5T^{2} \)
29 \( 1 + (-481. - 156. i)T + (5.72e5 + 4.15e5i)T^{2} \)
31 \( 1 + (-192. + 139. i)T + (2.85e5 - 8.78e5i)T^{2} \)
37 \( 1 + (718. - 2.21e3i)T + (-1.51e6 - 1.10e6i)T^{2} \)
41 \( 1 + (-100. + 32.8i)T + (2.28e6 - 1.66e6i)T^{2} \)
43 \( 1 + 2.05e3iT - 3.41e6T^{2} \)
47 \( 1 + (17.8 + 55.0i)T + (-3.94e6 + 2.86e6i)T^{2} \)
53 \( 1 + (926. - 673. i)T + (2.43e6 - 7.50e6i)T^{2} \)
59 \( 1 + (-596. + 1.83e3i)T + (-9.80e6 - 7.12e6i)T^{2} \)
61 \( 1 + (375. - 516. i)T + (-4.27e6 - 1.31e7i)T^{2} \)
67 \( 1 + 8.69e3T + 2.01e7T^{2} \)
71 \( 1 + (1.51e3 + 1.10e3i)T + (7.85e6 + 2.41e7i)T^{2} \)
73 \( 1 + (-2.33e3 - 758. i)T + (2.29e7 + 1.66e7i)T^{2} \)
79 \( 1 + (2.25e3 + 3.10e3i)T + (-1.20e7 + 3.70e7i)T^{2} \)
83 \( 1 + (4.35e3 - 5.98e3i)T + (-1.46e7 - 4.51e7i)T^{2} \)
89 \( 1 + 1.11e4T + 6.27e7T^{2} \)
97 \( 1 + (-9.94e3 + 7.22e3i)T + (2.73e7 - 8.41e7i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.79610679467467191088559131543, −12.15617493853987050078044485674, −11.56659158888029754834236728122, −10.03160572559233117616114015908, −8.607281261750552419954825605472, −7.54477327766632380704985378280, −6.40012216496832146735409944086, −5.00249445423545844754592889047, −4.16471457094095137294149132072, −1.50398964315719950634970886346, 1.40117731640940843744280793309, 3.30202502031077898934333818962, 4.16835415194534047875133297580, 5.82948600932646086912431817554, 7.65349680994360397871787739417, 8.359342882282993532076791470798, 10.40999758340146303661057923238, 11.06892367006116231535734491034, 11.87735926014596391843921779851, 12.88514062676020748405063810409

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