↑

Properties

Label 2-84-12.11-c1-0-7
Degree 22
Conductor 8484
Sign 0.998+0.0608i0.998 + 0.0608i
Analytic cond. 0.6707430.670743
Root an. cond. 0.8189890.818989
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  + (1.34 + 0.430i)2-s + (−0.916 − 1.46i)3-s + (1.62 + 1.15i)4-s + 0.348i·5-s + (−0.602 − 2.37i)6-s − i·7-s + (1.69 + 2.26i)8-s + (−1.31 + 2.69i)9-s + (−0.150 + 0.469i)10-s − 3.90·11-s + (0.210 − 3.45i)12-s − 2.93·13-s + (0.430 − 1.34i)14-s + (0.512 − 0.319i)15-s + (1.30 + 3.77i)16-s − 3.90i·17-s + ⋯
L(s)  = 1  + (0.952 + 0.304i)2-s + (−0.529 − 0.848i)3-s + (0.814 + 0.579i)4-s + 0.155i·5-s + (−0.245 − 0.969i)6-s − 0.377i·7-s + (0.599 + 0.800i)8-s + (−0.439 + 0.898i)9-s + (−0.0474 + 0.148i)10-s − 1.17·11-s + (0.0608 − 0.998i)12-s − 0.815·13-s + (0.115 − 0.360i)14-s + (0.132 − 0.0825i)15-s + (0.327 + 0.944i)16-s − 0.946i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.998+0.0608i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0608i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+1/2) L(s)=((0.998+0.0608i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0608i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.998+0.0608i0.998 + 0.0608i
Analytic conductor: 0.6707430.670743
Root analytic conductor: 0.8189890.818989
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ84(71,⋅)\chi_{84} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :1/2), 0.998+0.0608i)(2,\ 84,\ (\ :1/2),\ 0.998 + 0.0608i)

Particular Values

L(1)L(1) ≈\approx 1.31965−0.0402016i1.31965 - 0.0402016i
L(12)L(\frac12) ≈\approx 1.31965−0.0402016i1.31965 - 0.0402016i
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+(−1.34−0.430i)T 1 + (-1.34 - 0.430i)T
3 1+(0.916+1.46i)T 1 + (0.916 + 1.46i)T
7 1+iT 1 + iT
good5 1−0.348iT−5T2 1 - 0.348iT - 5T^{2}
11 1+3.90T+11T2 1 + 3.90T + 11T^{2}
13 1+2.93T+13T2 1 + 2.93T + 13T^{2}
17 1+3.90iT−17T2 1 + 3.90iT - 17T^{2}
19 1−5.57iT−19T2 1 - 5.57iT - 19T^{2}
23 1−2.18T+23T2 1 - 2.18T + 23T^{2}
29 1+9.75iT−29T2 1 + 9.75iT - 29T^{2}
31 1+2.63iT−31T2 1 + 2.63iT - 31T^{2}
37 1−0.639T+37T2 1 - 0.639T + 37T^{2}
41 1−7.57iT−41T2 1 - 7.57iT - 41T^{2}
43 1+2.51iT−43T2 1 + 2.51iT - 43T^{2}
47 1−4.36T+47T2 1 - 4.36T + 47T^{2}
53 1−1.72iT−53T2 1 - 1.72iT - 53T^{2}
59 1−8.24T+59T2 1 - 8.24T + 59T^{2}
61 1+14.0T+61T2 1 + 14.0T + 61T^{2}
67 1+0.639iT−67T2 1 + 0.639iT - 67T^{2}
71 1+11.9T+71T2 1 + 11.9T + 71T^{2}
73 1−7.87T+73T2 1 - 7.87T + 73T^{2}
79 1+4iT−79T2 1 + 4iT - 79T^{2}
83 1+8.94T+83T2 1 + 8.94T + 83T^{2}
89 1−10.5iT−89T2 1 - 10.5iT - 89T^{2}
97 1+2T+97T2 1 + 2T + 97T^{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

−14.02981973548596367676632714643, −13.22235648863063013993121403244, −12.33942199242103248595506112614, −11.38457465708374488297603935742, −10.25994119439235670400913770433, −7.956038828687021013273706169893, −7.24717638784428222116721073743, −5.94396641703700883856860150464, −4.78771429570704647593834017756, −2.62369144497836808755406868877, 2.96303196562227100648239238537, 4.71684099634878143413359447914, 5.47885126754850771673006861106, 6.98465057703508213337928389440, 8.955071294499425078885367380622, 10.36700447579834258745451868445, 10.98410854322699648999967949636, 12.27672108807410175720734681211, 12.98037529658177347876485456420, 14.47228109343457678568331218541

Graph of the ZZ-function along the critical line