↑

Properties

Label 2-87-29.16-c1-0-1
Degree 22
Conductor 8787
Sign 0.526+0.850i0.526 + 0.850i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.50 − 1.20i)2-s + (0.623 − 0.781i)3-s + (3.56 + 4.46i)4-s + (2.28 + 1.10i)5-s + (−2.50 + 1.20i)6-s + (0.527 − 0.661i)7-s + (−2.29 − 10.0i)8-s + (−0.222 − 0.974i)9-s + (−4.39 − 5.51i)10-s + (−0.279 + 1.22i)11-s + 5.71·12-s + (0.494 − 2.16i)13-s + (−2.11 + 1.02i)14-s + (2.28 − 1.10i)15-s + (−3.83 + 16.8i)16-s + 3.75·17-s + ⋯
L(s)  = 1  + (−1.76 − 0.852i)2-s + (0.359 − 0.451i)3-s + (1.78 + 2.23i)4-s + (1.02 + 0.492i)5-s + (−1.02 + 0.492i)6-s + (0.199 − 0.250i)7-s + (−0.812 − 3.55i)8-s + (−0.0741 − 0.324i)9-s + (−1.38 − 1.74i)10-s + (−0.0843 + 0.369i)11-s + 1.65·12-s + (0.137 − 0.600i)13-s + (−0.566 + 0.272i)14-s + (0.590 − 0.284i)15-s + (−0.959 + 4.20i)16-s + 0.910·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.526+0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.526+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.526+0.850i0.526 + 0.850i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(16,⋅)\chi_{87} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.526+0.850i)(2,\ 87,\ (\ :1/2),\ 0.526 + 0.850i)

Particular Values

L(1)L(1) ≈\approx 0.522023−0.290891i0.522023 - 0.290891i
L(12)L(\frac12) ≈\approx 0.522023−0.290891i0.522023 - 0.290891i
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+(−0.623+0.781i)T 1 + (-0.623 + 0.781i)T
29 1+(−4.24−3.31i)T 1 + (-4.24 - 3.31i)T
good2 1+(2.50+1.20i)T+(1.24+1.56i)T2 1 + (2.50 + 1.20i)T + (1.24 + 1.56i)T^{2}
5 1+(−2.28−1.10i)T+(3.11+3.90i)T2 1 + (-2.28 - 1.10i)T + (3.11 + 3.90i)T^{2}
7 1+(−0.527+0.661i)T+(−1.55−6.82i)T2 1 + (-0.527 + 0.661i)T + (-1.55 - 6.82i)T^{2}
11 1+(0.279−1.22i)T+(−9.91−4.77i)T2 1 + (0.279 - 1.22i)T + (-9.91 - 4.77i)T^{2}
13 1+(−0.494+2.16i)T+(−11.7−5.64i)T2 1 + (-0.494 + 2.16i)T + (-11.7 - 5.64i)T^{2}
17 1−3.75T+17T2 1 - 3.75T + 17T^{2}
19 1+(1.84+2.31i)T+(−4.22+18.5i)T2 1 + (1.84 + 2.31i)T + (-4.22 + 18.5i)T^{2}
23 1+(5.78−2.78i)T+(14.3−17.9i)T2 1 + (5.78 - 2.78i)T + (14.3 - 17.9i)T^{2}
31 1+(0.518+0.249i)T+(19.3+24.2i)T2 1 + (0.518 + 0.249i)T + (19.3 + 24.2i)T^{2}
37 1+(−0.893−3.91i)T+(−33.3+16.0i)T2 1 + (-0.893 - 3.91i)T + (-33.3 + 16.0i)T^{2}
41 1+9.29T+41T2 1 + 9.29T + 41T^{2}
43 1+(8.60−4.14i)T+(26.8−33.6i)T2 1 + (8.60 - 4.14i)T + (26.8 - 33.6i)T^{2}
47 1+(−0.626+2.74i)T+(−42.3−20.3i)T2 1 + (-0.626 + 2.74i)T + (-42.3 - 20.3i)T^{2}
53 1+(3.55+1.71i)T+(33.0+41.4i)T2 1 + (3.55 + 1.71i)T + (33.0 + 41.4i)T^{2}
59 1−2.24T+59T2 1 - 2.24T + 59T^{2}
61 1+(2.97−3.72i)T+(−13.5−59.4i)T2 1 + (2.97 - 3.72i)T + (-13.5 - 59.4i)T^{2}
67 1+(−0.0320−0.140i)T+(−60.3+29.0i)T2 1 + (-0.0320 - 0.140i)T + (-60.3 + 29.0i)T^{2}
71 1+(−1.13+4.99i)T+(−63.9−30.8i)T2 1 + (-1.13 + 4.99i)T + (-63.9 - 30.8i)T^{2}
73 1+(1.24−0.598i)T+(45.5−57.0i)T2 1 + (1.24 - 0.598i)T + (45.5 - 57.0i)T^{2}
79 1+(0.0599+0.262i)T+(−71.1+34.2i)T2 1 + (0.0599 + 0.262i)T + (-71.1 + 34.2i)T^{2}
83 1+(4.26+5.34i)T+(−18.4+80.9i)T2 1 + (4.26 + 5.34i)T + (-18.4 + 80.9i)T^{2}
89 1+(−4.15−2.00i)T+(55.4+69.5i)T2 1 + (-4.15 - 2.00i)T + (55.4 + 69.5i)T^{2}
97 1+(2.88+3.61i)T+(−21.5+94.5i)T2 1 + (2.88 + 3.61i)T + (-21.5 + 94.5i)T^{2}
show more
show less
   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

−13.77059105687630994354561604096, −12.65472005504019981324689488663, −11.60270799855220078889313773971, −10.29866760983284353268012804356, −9.875936978016397808563769683782, −8.558402936610543254441404295949, −7.59323603489120602776404489095, −6.43143307308723813462233884982, −3.09154966484573071098308588733, −1.71581533436500632589262428423, 1.91455088683630914974446783842, 5.39452862743073778796032762475, 6.40624947595454129014980299109, 8.033456228901254714643979793390, 8.778360091275642906152686775005, 9.782352911471439748439430531177, 10.38156616128725646636368702199, 11.81781348713223524384667220142, 13.86074442570908331572554390460, 14.63677874606697569757893119269

Graph of the ZZ-function along the critical line