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Properties

Label 2-99-11.8-c2-0-1
Degree $2$
Conductor $99$
Sign $-0.703 - 0.710i$
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  + (2.27 + 3.12i)2-s + (−3.37 + 10.3i)4-s + (−3.35 − 2.43i)5-s + (7.08 + 2.30i)7-s + (−25.4 + 8.26i)8-s − 16.0i·10-s + (8.33 − 7.17i)11-s + (4.30 + 5.92i)13-s + (8.88 + 27.3i)14-s + (−48.2 − 35.0i)16-s + (2.10 − 2.89i)17-s + (28.5 − 9.26i)19-s + (36.6 − 26.6i)20-s + (41.3 + 9.75i)22-s − 20.6·23-s + ⋯
L(s)  = 1  + (1.13 + 1.56i)2-s + (−0.843 + 2.59i)4-s + (−0.671 − 0.487i)5-s + (1.01 + 0.328i)7-s + (−3.18 + 1.03i)8-s − 1.60i·10-s + (0.757 − 0.652i)11-s + (0.331 + 0.455i)13-s + (0.634 + 1.95i)14-s + (−3.01 − 2.19i)16-s + (0.123 − 0.170i)17-s + (1.50 − 0.487i)19-s + (1.83 − 1.33i)20-s + (1.87 + 0.443i)22-s − 0.897·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.703 - 0.710i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.703 - 0.710i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.813827 + 1.95067i\)
\(L(\frac12)\) \(\approx\) \(0.813827 + 1.95067i\)
\(L(2)\) 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 \( 1 + (-8.33 + 7.17i)T \)
good2 \( 1 + (-2.27 - 3.12i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (3.35 + 2.43i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (-7.08 - 2.30i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-4.30 - 5.92i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-2.10 + 2.89i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-28.5 + 9.26i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 20.6T + 529T^{2} \)
29 \( 1 + (16.3 + 5.31i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (12.7 - 9.27i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (9.97 - 30.6i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-15.4 + 5.01i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 33.7iT - 1.84e3T^{2} \)
47 \( 1 + (12.0 + 36.9i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-29.6 + 21.5i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (21.6 - 66.5i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-13.3 + 18.4i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 63.0T + 4.48e3T^{2} \)
71 \( 1 + (25.5 + 18.5i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (98.3 + 31.9i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-10.7 - 14.7i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (69.9 - 96.2i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 154.T + 7.92e3T^{2} \)
97 \( 1 + (-55.9 + 40.6i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−14.14277258096324171761812663605, −13.47282397024192614933659954962, −11.94740576682504435471057386290, −11.69025292238409623898134556397, −8.988224326711723684254046907610, −8.200810656531396046634183896553, −7.22498898282732863827144574730, −5.84644575599796372317716493141, −4.78872919137005308977388597371, −3.67168345313694897115415754231, 1.54117178126884650027343973143, 3.41051205746438686708626275837, 4.39829381858545693864955249038, 5.73752902048146399765785900024, 7.58418365779920824127409406151, 9.419695472816049367943799672391, 10.52710253580581339562654941724, 11.42947474587192148679923438559, 11.95639679081726301142162674731, 13.09868221634234605697849984357

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