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Properties

Label 2-87-29.23-c1-0-1
Degree 22
Conductor 8787
Sign −0.243−0.969i-0.243 - 0.969i
Analytic cond. 0.6946980.694698
Root an. cond. 0.8334850.833485
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  + (−0.754 + 0.946i)2-s + (−0.222 + 0.974i)3-s + (0.118 + 0.520i)4-s + (1.12 − 1.40i)5-s + (−0.754 − 0.946i)6-s + (−0.951 + 4.16i)7-s + (−2.76 − 1.33i)8-s + (−0.900 − 0.433i)9-s + (0.484 + 2.12i)10-s + (−0.951 + 0.458i)11-s − 0.534·12-s + (4.75 − 2.29i)13-s + (−3.22 − 4.04i)14-s + (1.12 + 1.40i)15-s + (2.38 − 1.14i)16-s + 5.61·17-s + ⋯
L(s)  = 1  + (−0.533 + 0.669i)2-s + (−0.128 + 0.562i)3-s + (0.0594 + 0.260i)4-s + (0.501 − 0.628i)5-s + (−0.308 − 0.386i)6-s + (−0.359 + 1.57i)7-s + (−0.977 − 0.470i)8-s + (−0.300 − 0.144i)9-s + (0.153 + 0.671i)10-s + (−0.286 + 0.138i)11-s − 0.154·12-s + (1.31 − 0.635i)13-s + (−0.862 − 1.08i)14-s + (0.289 + 0.363i)15-s + (0.596 − 0.287i)16-s + 1.36·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((−0.243−0.969i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((−0.243−0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: −0.243−0.969i-0.243 - 0.969i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(52,⋅)\chi_{87} (52, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), −0.243−0.969i)(2,\ 87,\ (\ :1/2),\ -0.243 - 0.969i)

Particular Values

L(1)L(1) ≈\approx 0.469740+0.602017i0.469740 + 0.602017i
L(12)L(\frac12) ≈\approx 0.469740+0.602017i0.469740 + 0.602017i
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+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
29 1+(4.54+2.88i)T 1 + (4.54 + 2.88i)T
good2 1+(0.754−0.946i)T+(−0.445−1.94i)T2 1 + (0.754 - 0.946i)T + (-0.445 - 1.94i)T^{2}
5 1+(−1.12+1.40i)T+(−1.11−4.87i)T2 1 + (-1.12 + 1.40i)T + (-1.11 - 4.87i)T^{2}
7 1+(0.951−4.16i)T+(−6.30−3.03i)T2 1 + (0.951 - 4.16i)T + (-6.30 - 3.03i)T^{2}
11 1+(0.951−0.458i)T+(6.85−8.60i)T2 1 + (0.951 - 0.458i)T + (6.85 - 8.60i)T^{2}
13 1+(−4.75+2.29i)T+(8.10−10.1i)T2 1 + (-4.75 + 2.29i)T + (8.10 - 10.1i)T^{2}
17 1−5.61T+17T2 1 - 5.61T + 17T^{2}
19 1+(1.42+6.25i)T+(−17.1+8.24i)T2 1 + (1.42 + 6.25i)T + (-17.1 + 8.24i)T^{2}
23 1+(−1.27−1.59i)T+(−5.11+22.4i)T2 1 + (-1.27 - 1.59i)T + (-5.11 + 22.4i)T^{2}
31 1+(−2.38+2.99i)T+(−6.89−30.2i)T2 1 + (-2.38 + 2.99i)T + (-6.89 - 30.2i)T^{2}
37 1+(−0.315−0.151i)T+(23.0+28.9i)T2 1 + (-0.315 - 0.151i)T + (23.0 + 28.9i)T^{2}
41 1+3.62T+41T2 1 + 3.62T + 41T^{2}
43 1+(−1.09−1.36i)T+(−9.56+41.9i)T2 1 + (-1.09 - 1.36i)T + (-9.56 + 41.9i)T^{2}
47 1+(6.28−3.02i)T+(29.3−36.7i)T2 1 + (6.28 - 3.02i)T + (29.3 - 36.7i)T^{2}
53 1+(4.87−6.11i)T+(−11.7−51.6i)T2 1 + (4.87 - 6.11i)T + (-11.7 - 51.6i)T^{2}
59 1−0.382T+59T2 1 - 0.382T + 59T^{2}
61 1+(−1.21+5.32i)T+(−54.9−26.4i)T2 1 + (-1.21 + 5.32i)T + (-54.9 - 26.4i)T^{2}
67 1+(7.03+3.38i)T+(41.7+52.3i)T2 1 + (7.03 + 3.38i)T + (41.7 + 52.3i)T^{2}
71 1+(−14.0+6.77i)T+(44.2−55.5i)T2 1 + (-14.0 + 6.77i)T + (44.2 - 55.5i)T^{2}
73 1+(1.58+1.98i)T+(−16.2+71.1i)T2 1 + (1.58 + 1.98i)T + (-16.2 + 71.1i)T^{2}
79 1+(9.25+4.45i)T+(49.2+61.7i)T2 1 + (9.25 + 4.45i)T + (49.2 + 61.7i)T^{2}
83 1+(0.338+1.48i)T+(−74.7+36.0i)T2 1 + (0.338 + 1.48i)T + (-74.7 + 36.0i)T^{2}
89 1+(11.1−14.0i)T+(−19.8−86.7i)T2 1 + (11.1 - 14.0i)T + (-19.8 - 86.7i)T^{2}
97 1+(−1.51−6.63i)T+(−87.3+42.0i)T2 1 + (-1.51 - 6.63i)T + (-87.3 + 42.0i)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

−15.15152678896365793711870784157, −13.26551628657602676815675150162, −12.45953994059274232264798239126, −11.29993887468933255628308287625, −9.599277752725495073937866476344, −8.987138622015171827050720277254, −8.025347518846611524254159839140, −6.19504583360263124268831505385, −5.36399750697923939943838110292, −3.11329208469633293741539300682, 1.40518222025409769858113908078, 3.45775692869429023077446155565, 5.90358875668298089693931688396, 6.88335105188030357855003100319, 8.332294506402500083140509045419, 9.981543815074833337381663571605, 10.47934467393749412346609923507, 11.42246077178445240105968492306, 12.79094537909157446305172740334, 13.98071191794682076709509223026

Graph of the ZZ-function along the critical line