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Properties

Label 2-99-11.7-c2-0-8
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: $\chi_{99} (73, \cdot )$
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

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

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