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Properties

Label 2-99-11.3-c3-0-8
Degree 22
Conductor 9999
Sign 0.422+0.906i0.422 + 0.906i
Analytic cond. 5.841185.84118
Root an. cond. 2.416852.41685
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.523 + 0.380i)2-s + (−2.34 + 7.21i)4-s + (−9.01 − 6.54i)5-s + (8.07 − 24.8i)7-s + (−3.11 − 9.58i)8-s + 7.20·10-s + (36.0 − 5.31i)11-s + (43.2 − 31.4i)13-s + (5.22 + 16.0i)14-s + (−43.7 − 31.8i)16-s + (18.9 + 13.7i)17-s + (−21.8 − 67.3i)19-s + (68.3 − 49.6i)20-s + (−16.8 + 16.5i)22-s − 164.·23-s + ⋯
L(s)  = 1  + (−0.185 + 0.134i)2-s + (−0.292 + 0.901i)4-s + (−0.806 − 0.585i)5-s + (0.436 − 1.34i)7-s + (−0.137 − 0.423i)8-s + 0.227·10-s + (0.989 − 0.145i)11-s + (0.923 − 0.671i)13-s + (0.0997 + 0.307i)14-s + (−0.684 − 0.497i)16-s + (0.270 + 0.196i)17-s + (−0.264 − 0.813i)19-s + (0.764 − 0.555i)20-s + (−0.163 + 0.159i)22-s − 1.48·23-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((0.422+0.906i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+3/2) L(s)=((0.422+0.906i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 0.422+0.906i0.422 + 0.906i
Analytic conductor: 5.841185.84118
Root analytic conductor: 2.416852.41685
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ99(91,⋅)\chi_{99} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 99, ( :3/2), 0.422+0.906i)(2,\ 99,\ (\ :3/2),\ 0.422 + 0.906i)

Particular Values

L(2)L(2) ≈\approx 0.891548−0.567822i0.891548 - 0.567822i
L(12)L(\frac12) ≈\approx 0.891548−0.567822i0.891548 - 0.567822i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
11 1+(−36.0+5.31i)T 1 + (-36.0 + 5.31i)T
good2 1+(0.523−0.380i)T+(2.47−7.60i)T2 1 + (0.523 - 0.380i)T + (2.47 - 7.60i)T^{2}
5 1+(9.01+6.54i)T+(38.6+118.i)T2 1 + (9.01 + 6.54i)T + (38.6 + 118. i)T^{2}
7 1+(−8.07+24.8i)T+(−277.−201.i)T2 1 + (-8.07 + 24.8i)T + (-277. - 201. i)T^{2}
13 1+(−43.2+31.4i)T+(678.−2.08e3i)T2 1 + (-43.2 + 31.4i)T + (678. - 2.08e3i)T^{2}
17 1+(−18.9−13.7i)T+(1.51e3+4.67e3i)T2 1 + (-18.9 - 13.7i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(21.8+67.3i)T+(−5.54e3+4.03e3i)T2 1 + (21.8 + 67.3i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1+164.T+1.21e4T2 1 + 164.T + 1.21e4T^{2}
29 1+(−67.5+207.i)T+(−1.97e4−1.43e4i)T2 1 + (-67.5 + 207. i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−62.0+45.0i)T+(9.20e3−2.83e4i)T2 1 + (-62.0 + 45.0i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(87.5−269.i)T+(−4.09e4−2.97e4i)T2 1 + (87.5 - 269. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(1.50+4.61i)T+(−5.57e4+4.05e4i)T2 1 + (1.50 + 4.61i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+333.T+7.95e4T2 1 + 333.T + 7.95e4T^{2}
47 1+(−121.−374.i)T+(−8.39e4+6.10e4i)T2 1 + (-121. - 374. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(−123.+89.3i)T+(4.60e4−1.41e5i)T2 1 + (-123. + 89.3i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−237.+729.i)T+(−1.66e5−1.20e5i)T2 1 + (-237. + 729. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−287.−209.i)T+(7.01e4+2.15e5i)T2 1 + (-287. - 209. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−102.T+3.00e5T2 1 - 102.T + 3.00e5T^{2}
71 1+(−504.−366.i)T+(1.10e5+3.40e5i)T2 1 + (-504. - 366. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(95.7−294.i)T+(−3.14e5−2.28e5i)T2 1 + (95.7 - 294. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(−517.+375.i)T+(1.52e5−4.68e5i)T2 1 + (-517. + 375. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(−233.−169.i)T+(1.76e5+5.43e5i)T2 1 + (-233. - 169. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+184.T+7.04e5T2 1 + 184.T + 7.04e5T^{2}
97 1+(515.−374.i)T+(2.82e5−8.68e5i)T2 1 + (515. - 374. i)T + (2.82e5 - 8.68e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.28724106722898979139027883926, −12.10151423936794699998314980753, −11.30349495104718837014735216023, −9.900411016062100504856337676633, −8.368351166543914633221355405535, −7.970116669740604855620791670873, −6.60867442415890543126393879583, −4.40684819299942353704182234403, −3.72079209624106745531914731642, −0.69565277785851343745301982854, 1.78762179537612582089628097815, 3.88800350485130563501896733421, 5.52186835758060432085235190682, 6.63192604595208001595682756831, 8.353457869854073644136944264129, 9.153814683935868282834850639294, 10.45672313848952541301357190913, 11.56604369330660049035750173033, 12.11963172960743263817263301555, 13.96730565969226906137686398167

Graph of the ZZ-function along the critical line