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Properties

Label 2-87-29.28-c1-0-0
Degree 22
Conductor 8787
Sign 0.557−0.830i0.557 - 0.830i
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  + 0.618i·2-s + i·3-s + 1.61·4-s − 1.23·5-s − 0.618·6-s + 0.236·7-s + 2.23i·8-s − 9-s − 0.763i·10-s − 2.23i·11-s + 1.61i·12-s − 13-s + 0.145i·14-s − 1.23i·15-s + 1.85·16-s − 3i·17-s + ⋯
L(s)  = 1  + 0.437i·2-s + 0.577i·3-s + 0.809·4-s − 0.552·5-s − 0.252·6-s + 0.0892·7-s + 0.790i·8-s − 0.333·9-s − 0.241i·10-s − 0.674i·11-s + 0.467i·12-s − 0.277·13-s + 0.0389i·14-s − 0.319i·15-s + 0.463·16-s − 0.727i·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.557−0.830i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.557−0.830i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.557−0.830i0.557 - 0.830i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(28,⋅)\chi_{87} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.557−0.830i)(2,\ 87,\ (\ :1/2),\ 0.557 - 0.830i)

Particular Values

L(1)L(1) ≈\approx 0.924735+0.493197i0.924735 + 0.493197i
L(12)L(\frac12) ≈\approx 0.924735+0.493197i0.924735 + 0.493197i
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−iT 1 - iT
29 1+(−4.47−3i)T 1 + (-4.47 - 3i)T
good2 1−0.618iT−2T2 1 - 0.618iT - 2T^{2}
5 1+1.23T+5T2 1 + 1.23T + 5T^{2}
7 1−0.236T+7T2 1 - 0.236T + 7T^{2}
11 1+2.23iT−11T2 1 + 2.23iT - 11T^{2}
13 1+T+13T2 1 + T + 13T^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1+5.70iT−19T2 1 + 5.70iT - 19T^{2}
23 1+3.23T+23T2 1 + 3.23T + 23T^{2}
31 1+2.76iT−31T2 1 + 2.76iT - 31T^{2}
37 1−9.23iT−37T2 1 - 9.23iT - 37T^{2}
41 1−4.47iT−41T2 1 - 4.47iT - 41T^{2}
43 1−8.47iT−43T2 1 - 8.47iT - 43T^{2}
47 1+5.76iT−47T2 1 + 5.76iT - 47T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1+8.94T+59T2 1 + 8.94T + 59T^{2}
61 1−7.23iT−61T2 1 - 7.23iT - 61T^{2}
67 1+8.70T+67T2 1 + 8.70T + 67T^{2}
71 1−9.23T+71T2 1 - 9.23T + 71T^{2}
73 1−5.70iT−73T2 1 - 5.70iT - 73T^{2}
79 1+15.7iT−79T2 1 + 15.7iT - 79T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+17.9iT−89T2 1 + 17.9iT - 89T^{2}
97 1+4.18iT−97T2 1 + 4.18iT - 97T^{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

−14.67096616241862822894929310969, −13.52028951706159641456029378348, −11.81541397245759299695200739327, −11.32242339419112691341998908021, −10.07391762188405516333126232330, −8.616094809307978180108285806494, −7.52319752337265024985672880202, −6.25210055772379311908766450576, −4.80171995312976610053413441723, −2.97038936806824430773598937284, 2.00718633194260351255256283593, 3.84268687105472536740287733842, 5.96468628375135172750882048602, 7.23660794278294829129180385198, 8.132810516947386341120299804898, 9.908678994501687779211182638847, 10.92493536013980766844073253122, 12.21812210801941119359865164619, 12.37674484358416324759885608696, 14.00309307262373821971562443762

Graph of the ZZ-function along the critical line