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Properties

Label 2-99-9.4-c1-0-6
Degree 22
Conductor 9999
Sign 0.985−0.172i0.985 - 0.172i
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  + (0.447 + 0.774i)2-s + (1.22 − 1.22i)3-s + (0.599 − 1.03i)4-s + (−1.87 + 3.24i)5-s + (1.49 + 0.401i)6-s + (−0.725 − 1.25i)7-s + 2.86·8-s + (0.00384 − 2.99i)9-s − 3.35·10-s + (−0.5 − 0.866i)11-s + (−0.536 − 2.00i)12-s + (−2.87 + 4.98i)13-s + (0.648 − 1.12i)14-s + (1.67 + 6.27i)15-s + (0.0800 + 0.138i)16-s − 4.79·17-s + ⋯
L(s)  = 1  + (0.316 + 0.547i)2-s + (0.707 − 0.706i)3-s + (0.299 − 0.519i)4-s + (−0.838 + 1.45i)5-s + (0.610 + 0.164i)6-s + (−0.274 − 0.474i)7-s + 1.01·8-s + (0.00128 − 0.999i)9-s − 1.06·10-s + (−0.150 − 0.261i)11-s + (−0.154 − 0.579i)12-s + (−0.798 + 1.38i)13-s + (0.173 − 0.300i)14-s + (0.432 + 1.61i)15-s + (0.0200 + 0.0346i)16-s − 1.16·17-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((0.985−0.172i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+1/2) L(s)=((0.985−0.172i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 0.985−0.172i0.985 - 0.172i
Analytic conductor: 0.7905180.790518
Root analytic conductor: 0.8891110.889111
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ99(67,⋅)\chi_{99} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 99, ( :1/2), 0.985−0.172i)(2,\ 99,\ (\ :1/2),\ 0.985 - 0.172i)

Particular Values

L(1)L(1) ≈\approx 1.31630+0.114312i1.31630 + 0.114312i
L(12)L(\frac12) ≈\approx 1.31630+0.114312i1.31630 + 0.114312i
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+(−1.22+1.22i)T 1 + (-1.22 + 1.22i)T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(−0.447−0.774i)T+(−1+1.73i)T2 1 + (-0.447 - 0.774i)T + (-1 + 1.73i)T^{2}
5 1+(1.87−3.24i)T+(−2.5−4.33i)T2 1 + (1.87 - 3.24i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.725+1.25i)T+(−3.5+6.06i)T2 1 + (0.725 + 1.25i)T + (-3.5 + 6.06i)T^{2}
13 1+(2.87−4.98i)T+(−6.5−11.2i)T2 1 + (2.87 - 4.98i)T + (-6.5 - 11.2i)T^{2}
17 1+4.79T+17T2 1 + 4.79T + 17T^{2}
19 1−0.702T+19T2 1 - 0.702T + 19T^{2}
23 1+(−0.825+1.42i)T+(−11.5−19.9i)T2 1 + (-0.825 + 1.42i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.15−3.72i)T+(−14.5+25.1i)T2 1 + (-2.15 - 3.72i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.65+2.86i)T+(−15.5−26.8i)T2 1 + (-1.65 + 2.86i)T + (-15.5 - 26.8i)T^{2}
37 1−9.73T+37T2 1 - 9.73T + 37T^{2}
41 1+(2.12−3.67i)T+(−20.5−35.5i)T2 1 + (2.12 - 3.67i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.05−3.55i)T+(−21.5+37.2i)T2 1 + (-2.05 - 3.55i)T + (-21.5 + 37.2i)T^{2}
47 1+(0.898+1.55i)T+(−23.5+40.7i)T2 1 + (0.898 + 1.55i)T + (-23.5 + 40.7i)T^{2}
53 1+1.15T+53T2 1 + 1.15T + 53T^{2}
59 1+(−2.32+4.02i)T+(−29.5−51.0i)T2 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.27+2.20i)T+(−30.5+52.8i)T2 1 + (1.27 + 2.20i)T + (-30.5 + 52.8i)T^{2}
67 1+(4.47−7.74i)T+(−33.5−58.0i)T2 1 + (4.47 - 7.74i)T + (-33.5 - 58.0i)T^{2}
71 1−5.14T+71T2 1 - 5.14T + 71T^{2}
73 1−10.5T+73T2 1 - 10.5T + 73T^{2}
79 1+(−0.543−0.941i)T+(−39.5+68.4i)T2 1 + (-0.543 - 0.941i)T + (-39.5 + 68.4i)T^{2}
83 1+(−1.90−3.29i)T+(−41.5+71.8i)T2 1 + (-1.90 - 3.29i)T + (-41.5 + 71.8i)T^{2}
89 1+4.01T+89T2 1 + 4.01T + 89T^{2}
97 1+(1.64+2.85i)T+(−48.5+84.0i)T2 1 + (1.64 + 2.85i)T + (-48.5 + 84.0i)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

−14.22609111974396112909102139623, −13.30563508595311377656147436803, −11.71717730941117642097035276205, −10.89851348719835940624499416111, −9.612619986259039467935072649630, −7.912919709698364022535120311952, −6.87426322366561269074669448843, −6.58999612750236329830392779345, −4.20396939852830988930132799869, −2.53264539477607638353543442603, 2.68795680624850277873737133833, 4.13303628190339915877173735442, 5.07094737039958634117753956355, 7.63944406405573174015519526584, 8.374108585475706663460734330347, 9.463464412505869606654904543640, 10.78875164902541139869842643641, 12.00881297090682081789954659508, 12.74395279961215987922752073318, 13.48744339972181298404034992029

Graph of the ZZ-function along the critical line