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Properties

Label 2-99-11.4-c1-0-3
Degree 22
Conductor 9999
Sign 0.530+0.847i0.530 + 0.847i
Analytic cond. 0.7905180.790518
Root an. cond. 0.8891110.889111
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.224i)2-s + (−0.572 − 1.76i)4-s + (1.30 − 0.951i)5-s + (−0.309 − 0.951i)7-s + (−0.454 + 1.40i)8-s − 0.618·10-s + (2.19 − 2.48i)11-s + (3.42 + 2.48i)13-s + (−0.118 + 0.363i)14-s + (−2.54 + 1.84i)16-s + (−6.35 + 4.61i)17-s + (−0.263 + 0.812i)19-s + (−2.42 − 1.76i)20-s + (−1.23 + 0.277i)22-s + 4.23·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.158i)2-s + (−0.286 − 0.881i)4-s + (0.585 − 0.425i)5-s + (−0.116 − 0.359i)7-s + (−0.160 + 0.495i)8-s − 0.195·10-s + (0.660 − 0.750i)11-s + (0.950 + 0.690i)13-s + (−0.0315 + 0.0970i)14-s + (−0.636 + 0.462i)16-s + (−1.54 + 1.11i)17-s + (−0.0605 + 0.186i)19-s + (−0.542 − 0.394i)20-s + (−0.263 + 0.0591i)22-s + 0.883·23-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((0.530+0.847i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+1/2) L(s)=((0.530+0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 0.530+0.847i0.530 + 0.847i
Analytic conductor: 0.7905180.790518
Root analytic conductor: 0.8891110.889111
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ99(37,⋅)\chi_{99} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 99, ( :1/2), 0.530+0.847i)(2,\ 99,\ (\ :1/2),\ 0.530 + 0.847i)

Particular Values

L(1)L(1) ≈\approx 0.809464−0.448545i0.809464 - 0.448545i
L(12)L(\frac12) ≈\approx 0.809464−0.448545i0.809464 - 0.448545i
L(32)L(\frac{3}{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+(−2.19+2.48i)T 1 + (-2.19 + 2.48i)T
good2 1+(0.309+0.224i)T+(0.618+1.90i)T2 1 + (0.309 + 0.224i)T + (0.618 + 1.90i)T^{2}
5 1+(−1.30+0.951i)T+(1.54−4.75i)T2 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2}
7 1+(0.309+0.951i)T+(−5.66+4.11i)T2 1 + (0.309 + 0.951i)T + (-5.66 + 4.11i)T^{2}
13 1+(−3.42−2.48i)T+(4.01+12.3i)T2 1 + (-3.42 - 2.48i)T + (4.01 + 12.3i)T^{2}
17 1+(6.35−4.61i)T+(5.25−16.1i)T2 1 + (6.35 - 4.61i)T + (5.25 - 16.1i)T^{2}
19 1+(0.263−0.812i)T+(−15.3−11.1i)T2 1 + (0.263 - 0.812i)T + (-15.3 - 11.1i)T^{2}
23 1−4.23T+23T2 1 - 4.23T + 23T^{2}
29 1+(−1.85−5.70i)T+(−23.4+17.0i)T2 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2}
31 1+(4.11+2.99i)T+(9.57+29.4i)T2 1 + (4.11 + 2.99i)T + (9.57 + 29.4i)T^{2}
37 1+(0.545+1.67i)T+(−29.9+21.7i)T2 1 + (0.545 + 1.67i)T + (-29.9 + 21.7i)T^{2}
41 1+(−1.30+4.02i)T+(−33.1−24.0i)T2 1 + (-1.30 + 4.02i)T + (-33.1 - 24.0i)T^{2}
43 1−6.70T+43T2 1 - 6.70T + 43T^{2}
47 1+(0.336−1.03i)T+(−38.0−27.6i)T2 1 + (0.336 - 1.03i)T + (-38.0 - 27.6i)T^{2}
53 1+(2.11+1.53i)T+(16.3+50.4i)T2 1 + (2.11 + 1.53i)T + (16.3 + 50.4i)T^{2}
59 1+(2.97+9.14i)T+(−47.7+34.6i)T2 1 + (2.97 + 9.14i)T + (-47.7 + 34.6i)T^{2}
61 1+(6.92−5.03i)T+(18.8−58.0i)T2 1 + (6.92 - 5.03i)T + (18.8 - 58.0i)T^{2}
67 1+4.85T+67T2 1 + 4.85T + 67T^{2}
71 1+(4.30−3.13i)T+(21.9−67.5i)T2 1 + (4.30 - 3.13i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.38−7.33i)T+(−59.0+42.9i)T2 1 + (-2.38 - 7.33i)T + (-59.0 + 42.9i)T^{2}
79 1+(8.89+6.46i)T+(24.4+75.1i)T2 1 + (8.89 + 6.46i)T + (24.4 + 75.1i)T^{2}
83 1+(6.04−4.39i)T+(25.6−78.9i)T2 1 + (6.04 - 4.39i)T + (25.6 - 78.9i)T^{2}
89 1−3.76T+89T2 1 - 3.76T + 89T^{2}
97 1+(0.927+0.673i)T+(29.9+92.2i)T2 1 + (0.927 + 0.673i)T + (29.9 + 92.2i)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

−13.70470492580803328094639729031, −12.97189447599362422994810679148, −11.23293497331166116966124842986, −10.64593736831630152959543672430, −9.159105410768445855660326247727, −8.783123492788275939600923885725, −6.64230974815025429215943007626, −5.68557377996537153024881361506, −4.12309431720945309107310287530, −1.54716083591508772365809195642, 2.75797128229835304904206030614, 4.44893809068220476898388007449, 6.27743133691307765662514150910, 7.31212599076008857144553093446, 8.748218401930449856053425731779, 9.475806944679677370225573928854, 10.90290839657879756723029561217, 12.04727202776054269509203972594, 13.12589318711766496610486109219, 13.85404142267583260855248494458

Graph of the ZZ-function along the critical line