↑

Properties

Label 2-862-431.10-c1-0-18
Degree 22
Conductor 862862
Sign 0.999+0.0206i0.999 + 0.0206i
Analytic cond. 6.883106.88310
Root an. cond. 2.623562.62356
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  + (0.109 − 0.994i)2-s + (2.20 − 0.323i)3-s + (−0.976 − 0.217i)4-s + (−2.19 + 0.0964i)5-s + (−0.0812 − 2.22i)6-s + (2.45 + 3.54i)7-s + (−0.322 + 0.946i)8-s + (1.86 − 0.562i)9-s + (−0.144 + 2.19i)10-s + (1.87 + 2.88i)11-s + (−2.21 − 0.162i)12-s + (1.27 + 1.40i)13-s + (3.79 − 2.05i)14-s + (−4.81 + 0.925i)15-s + (0.905 + 0.424i)16-s + (0.0570 − 1.11i)17-s + ⋯
L(s)  = 1  + (0.0773 − 0.702i)2-s + (1.27 − 0.187i)3-s + (−0.488 − 0.108i)4-s + (−0.983 + 0.0431i)5-s + (−0.0331 − 0.907i)6-s + (0.928 + 1.33i)7-s + (−0.114 + 0.334i)8-s + (0.622 − 0.187i)9-s + (−0.0457 + 0.694i)10-s + (0.566 + 0.870i)11-s + (−0.640 − 0.0468i)12-s + (0.354 + 0.390i)13-s + (1.01 − 0.549i)14-s + (−1.24 + 0.238i)15-s + (0.226 + 0.106i)16-s + (0.0138 − 0.270i)17-s + ⋯

Functional equation

Λ(s)=(862s/2 ΓC(s) L(s)=((0.999+0.0206i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(862s/2 ΓC(s+1/2) L(s)=((0.999+0.0206i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 862862    =    2⋅4312 \cdot 431
Sign: 0.999+0.0206i0.999 + 0.0206i
Analytic conductor: 6.883106.88310
Root analytic conductor: 2.623562.62356
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ862(441,⋅)\chi_{862} (441, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 862, ( :1/2), 0.999+0.0206i)(2,\ 862,\ (\ :1/2),\ 0.999 + 0.0206i)

Particular Values

L(1)L(1) ≈\approx 2.18184−0.0225836i2.18184 - 0.0225836i
L(12)L(\frac12) ≈\approx 2.18184−0.0225836i2.18184 - 0.0225836i
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)
bad2 1+(−0.109+0.994i)T 1 + (-0.109 + 0.994i)T
431 1+(−13.7+15.5i)T 1 + (-13.7 + 15.5i)T
good3 1+(−2.20+0.323i)T+(2.87−0.864i)T2 1 + (-2.20 + 0.323i)T + (2.87 - 0.864i)T^{2}
5 1+(2.19−0.0964i)T+(4.98−0.437i)T2 1 + (2.19 - 0.0964i)T + (4.98 - 0.437i)T^{2}
7 1+(−2.45−3.54i)T+(−2.45+6.55i)T2 1 + (-2.45 - 3.54i)T + (-2.45 + 6.55i)T^{2}
11 1+(−1.87−2.88i)T+(−4.44+10.0i)T2 1 + (-1.87 - 2.88i)T + (-4.44 + 10.0i)T^{2}
13 1+(−1.27−1.40i)T+(−1.23+12.9i)T2 1 + (-1.27 - 1.40i)T + (-1.23 + 12.9i)T^{2}
17 1+(−0.0570+1.11i)T+(−16.9−1.73i)T2 1 + (-0.0570 + 1.11i)T + (-16.9 - 1.73i)T^{2}
19 1+(−0.412−1.48i)T+(−16.2+9.77i)T2 1 + (-0.412 - 1.48i)T + (-16.2 + 9.77i)T^{2}
23 1+(−0.609−0.145i)T+(20.5+10.3i)T2 1 + (-0.609 - 0.145i)T + (20.5 + 10.3i)T^{2}
29 1+(−3.86−1.67i)T+(19.8+21.1i)T2 1 + (-3.86 - 1.67i)T + (19.8 + 21.1i)T^{2}
31 1+(0.248+0.0907i)T+(23.6+20.0i)T2 1 + (0.248 + 0.0907i)T + (23.6 + 20.0i)T^{2}
37 1+(−1.81−1.72i)T+(1.89+36.9i)T2 1 + (-1.81 - 1.72i)T + (1.89 + 36.9i)T^{2}
41 1+(−1.38+1.47i)T+(−2.69−40.9i)T2 1 + (-1.38 + 1.47i)T + (-2.69 - 40.9i)T^{2}
43 1+(0.250+3.80i)T+(−42.6+5.63i)T2 1 + (0.250 + 3.80i)T + (-42.6 + 5.63i)T^{2}
47 1+(−7.35+10.2i)T+(−15.1−44.4i)T2 1 + (-7.35 + 10.2i)T + (-15.1 - 44.4i)T^{2}
53 1+(−3.48+4.72i)T+(−15.6−50.6i)T2 1 + (-3.48 + 4.72i)T + (-15.6 - 50.6i)T^{2}
59 1+(−2.74−1.23i)T+(39.0+44.2i)T2 1 + (-2.74 - 1.23i)T + (39.0 + 44.2i)T^{2}
61 1+(−3.05+2.81i)T+(4.89−60.8i)T2 1 + (-3.05 + 2.81i)T + (4.89 - 60.8i)T^{2}
67 1+(1.33−2.80i)T+(−42.1−52.1i)T2 1 + (1.33 - 2.80i)T + (-42.1 - 52.1i)T^{2}
71 1+(0.0866+11.8i)T+(−70.9+1.03i)T2 1 + (0.0866 + 11.8i)T + (-70.9 + 1.03i)T^{2}
73 1+(3.62−2.32i)T+(30.5−66.3i)T2 1 + (3.62 - 2.32i)T + (30.5 - 66.3i)T^{2}
79 1+(10.4−0.304i)T+(78.8−4.61i)T2 1 + (10.4 - 0.304i)T + (78.8 - 4.61i)T^{2}
83 1+(11.9+6.25i)T+(47.3+68.2i)T2 1 + (11.9 + 6.25i)T + (47.3 + 68.2i)T^{2}
89 1+(4.05−1.97i)T+(54.9−70.0i)T2 1 + (4.05 - 1.97i)T + (54.9 - 70.0i)T^{2}
97 1+(−3.31−11.3i)T+(−81.6+52.3i)T2 1 + (-3.31 - 11.3i)T + (-81.6 + 52.3i)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

−10.03687319752332582494709177188, −9.048298009036441023443478585141, −8.636142767310613746454519791740, −7.945803548304898950797989760164, −7.05539652791007699162961321122, −5.54395371024328923137812986308, −4.45328039565324322943335318954, −3.61501420949203624633479029325, −2.51928379303508200857770383333, −1.70711338015244271834232614194, 1.00302175590974167228733510604, 3.02041731811293387111646769449, 4.00943454782589983993893532411, 4.37223103737386740155331964742, 5.87452623465266255596035306259, 7.13290597127850068539958232179, 7.78317094878769698629009082520, 8.307923238028414882760125368184, 8.916504197213773294515848922469, 10.01635517598144364027225620201

Graph of the ZZ-function along the critical line