↑

Properties

Label 2-980-140.139-c1-0-77
Degree 22
Conductor 980980
Sign 0.982−0.185i0.982 - 0.185i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (1.36 + 0.385i)2-s + 0.423i·3-s + (1.70 + 1.04i)4-s + (−1.74 − 1.39i)5-s + (−0.163 + 0.576i)6-s + (1.91 + 2.08i)8-s + 2.82·9-s + (−1.83 − 2.57i)10-s − 4.89i·11-s + (−0.443 + 0.721i)12-s − 2.54·13-s + (0.591 − 0.738i)15-s + (1.80 + 3.57i)16-s + 5.11·17-s + (3.83 + 1.08i)18-s + 6.26·19-s + ⋯
L(s)  = 1  + (0.962 + 0.272i)2-s + 0.244i·3-s + (0.851 + 0.524i)4-s + (−0.780 − 0.625i)5-s + (−0.0665 + 0.235i)6-s + (0.676 + 0.736i)8-s + 0.940·9-s + (−0.580 − 0.814i)10-s − 1.47i·11-s + (−0.128 + 0.208i)12-s − 0.706·13-s + (0.152 − 0.190i)15-s + (0.450 + 0.892i)16-s + 1.24·17-s + (0.904 + 0.256i)18-s + 1.43·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.982−0.185i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.982−0.185i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.982−0.185i0.982 - 0.185i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(979,⋅)\chi_{980} (979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.982−0.185i)(2,\ 980,\ (\ :1/2),\ 0.982 - 0.185i)

Particular Values

L(1)L(1) ≈\approx 2.84961+0.266501i2.84961 + 0.266501i
L(12)L(\frac12) ≈\approx 2.84961+0.266501i2.84961 + 0.266501i
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+(−1.36−0.385i)T 1 + (-1.36 - 0.385i)T
5 1+(1.74+1.39i)T 1 + (1.74 + 1.39i)T
7 1 1
good3 1−0.423iT−3T2 1 - 0.423iT - 3T^{2}
11 1+4.89iT−11T2 1 + 4.89iT - 11T^{2}
13 1+2.54T+13T2 1 + 2.54T + 13T^{2}
17 1−5.11T+17T2 1 - 5.11T + 17T^{2}
19 1−6.26T+19T2 1 - 6.26T + 19T^{2}
23 1−4.63T+23T2 1 - 4.63T + 23T^{2}
29 1+1.88T+29T2 1 + 1.88T + 29T^{2}
31 1+1.47T+31T2 1 + 1.47T + 31T^{2}
37 1−2.35iT−37T2 1 - 2.35iT - 37T^{2}
41 1−7.05iT−41T2 1 - 7.05iT - 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1+12.2iT−47T2 1 + 12.2iT - 47T^{2}
53 1−2.23iT−53T2 1 - 2.23iT - 53T^{2}
59 1+5.68T+59T2 1 + 5.68T + 59T^{2}
61 1+3.87iT−61T2 1 + 3.87iT - 61T^{2}
67 1−0.889T+67T2 1 - 0.889T + 67T^{2}
71 1+14.3iT−71T2 1 + 14.3iT - 71T^{2}
73 1−7.87T+73T2 1 - 7.87T + 73T^{2}
79 1+4.63iT−79T2 1 + 4.63iT - 79T^{2}
83 1−4.32iT−83T2 1 - 4.32iT - 83T^{2}
89 1−2.18iT−89T2 1 - 2.18iT - 89T^{2}
97 1+3.42T+97T2 1 + 3.42T + 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

−10.11066815469470715992608918168, −9.130871308074625656269509215038, −8.031058072387693213642777368197, −7.54753864409713299074087671294, −6.59323401952225004335020106978, −5.28949150664468443937013209439, −4.97831714020772332360148527307, −3.65878499240447562835692959285, −3.19986343888133786327154414798, −1.22356323714725716249020534757, 1.42213792312104382060461752986, 2.72424389259479294529683873622, 3.69612569081898843190419367685, 4.62461361777553207130721251226, 5.40688451665530415184873858863, 6.81390723619126144550400083089, 7.28810476159053368743458749645, 7.74613637623147234393259761646, 9.621233739597246051827154236759, 10.00534428435882195404308186025

Graph of the ZZ-function along the critical line