Properties

Label 2-99-33.8-c7-0-2
Degree $2$
Conductor $99$
Sign $-0.552 + 0.833i$
Analytic cond. $30.9261$
Root an. cond. $5.56112$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.5 + 12.7i)2-s + (105. − 324. i)4-s + (−159. + 218. i)5-s + (−1.13e3 − 369. i)7-s + (1.43e3 + 4.40e3i)8-s − 5.86e3i·10-s + (4.21e3 − 1.30e3i)11-s + (6.83e3 + 9.40e3i)13-s + (2.46e4 − 8.00e3i)14-s + (−4.57e4 − 3.32e4i)16-s + (−4.81e3 − 3.49e3i)17-s + (2.90e4 − 9.44e3i)19-s + (5.43e4 + 7.47e4i)20-s + (−5.72e4 + 7.66e4i)22-s + 9.65e4i·23-s + ⋯
L(s)  = 1  + (−1.54 + 1.12i)2-s + (0.824 − 2.53i)4-s + (−0.569 + 0.783i)5-s + (−1.25 − 0.406i)7-s + (0.987 + 3.03i)8-s − 1.85i·10-s + (0.955 − 0.296i)11-s + (0.862 + 1.18i)13-s + (2.39 − 0.779i)14-s + (−2.79 − 2.02i)16-s + (−0.237 − 0.172i)17-s + (0.971 − 0.315i)19-s + (1.51 + 2.09i)20-s + (−1.14 + 1.53i)22-s + 1.65i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(30.9261\)
Root analytic conductor: \(5.56112\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :7/2),\ -0.552 + 0.833i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.113205 - 0.210769i\)
\(L(\frac12)\) \(\approx\) \(0.113205 - 0.210769i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-4.21e3 + 1.30e3i)T \)
good2 \( 1 + (17.5 - 12.7i)T + (39.5 - 121. i)T^{2} \)
5 \( 1 + (159. - 218. i)T + (-2.41e4 - 7.43e4i)T^{2} \)
7 \( 1 + (1.13e3 + 369. i)T + (6.66e5 + 4.84e5i)T^{2} \)
13 \( 1 + (-6.83e3 - 9.40e3i)T + (-1.93e7 + 5.96e7i)T^{2} \)
17 \( 1 + (4.81e3 + 3.49e3i)T + (1.26e8 + 3.90e8i)T^{2} \)
19 \( 1 + (-2.90e4 + 9.44e3i)T + (7.23e8 - 5.25e8i)T^{2} \)
23 \( 1 - 9.65e4iT - 3.40e9T^{2} \)
29 \( 1 + (2.40e4 - 7.39e4i)T + (-1.39e10 - 1.01e10i)T^{2} \)
31 \( 1 + (1.27e5 - 9.24e4i)T + (8.50e9 - 2.61e10i)T^{2} \)
37 \( 1 + (-1.35e5 + 4.17e5i)T + (-7.68e10 - 5.57e10i)T^{2} \)
41 \( 1 + (7.87e3 + 2.42e4i)T + (-1.57e11 + 1.14e11i)T^{2} \)
43 \( 1 + 4.49e4iT - 2.71e11T^{2} \)
47 \( 1 + (-2.08e5 + 6.78e4i)T + (4.09e11 - 2.97e11i)T^{2} \)
53 \( 1 + (-4.87e5 - 6.70e5i)T + (-3.63e11 + 1.11e12i)T^{2} \)
59 \( 1 + (1.99e6 + 6.47e5i)T + (2.01e12 + 1.46e12i)T^{2} \)
61 \( 1 + (-5.00e5 + 6.88e5i)T + (-9.71e11 - 2.98e12i)T^{2} \)
67 \( 1 + 2.90e6T + 6.06e12T^{2} \)
71 \( 1 + (2.16e6 - 2.97e6i)T + (-2.81e12 - 8.64e12i)T^{2} \)
73 \( 1 + (-5.27e6 - 1.71e6i)T + (8.93e12 + 6.49e12i)T^{2} \)
79 \( 1 + (4.80e6 + 6.61e6i)T + (-5.93e12 + 1.82e13i)T^{2} \)
83 \( 1 + (-7.07e5 - 5.14e5i)T + (8.38e12 + 2.58e13i)T^{2} \)
89 \( 1 + 3.06e6iT - 4.42e13T^{2} \)
97 \( 1 + (2.33e6 - 1.69e6i)T + (2.49e13 - 7.68e13i)T^{2} \)
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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

−13.66974517430166124207615207434, −11.54346559234195776595582915514, −10.81405748818922146604854908053, −9.468398307231964301410818330313, −9.039516062400349878272690719018, −7.35682287608675723984663452146, −6.90590118024345476515842337628, −5.89427649978652172275054725155, −3.55936723267463279927116353501, −1.31471030181094250393703100760, 0.15961815634489932509973379907, 1.12382560307438628053844880763, 2.85218755554143003887214421009, 3.94841721143948741654995234902, 6.41904675917135222142081469047, 7.894486494638760371394229046563, 8.766154870438039058675836451764, 9.568302494181461080013963530255, 10.53028884721574670679636333951, 11.76792805023365423692904860809

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