↑

Properties

Label 2-80-16.13-c1-0-5
Degree 22
Conductor 8080
Sign 0.686+0.727i0.686 + 0.727i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
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  + (1.09 − 0.889i)2-s + (−0.120 − 0.120i)3-s + (0.418 − 1.95i)4-s + (−0.707 + 0.707i)5-s + (−0.238 − 0.0252i)6-s + 2.66i·7-s + (−1.27 − 2.52i)8-s − 2.97i·9-s + (−0.148 + 1.40i)10-s + (−3.49 + 3.49i)11-s + (−0.284 + 0.184i)12-s + (2.94 + 2.94i)13-s + (2.37 + 2.93i)14-s + 0.169·15-s + (−3.64 − 1.63i)16-s + 1.85·17-s + ⋯
L(s)  = 1  + (0.777 − 0.628i)2-s + (−0.0692 − 0.0692i)3-s + (0.209 − 0.977i)4-s + (−0.316 + 0.316i)5-s + (−0.0974 − 0.0103i)6-s + 1.00i·7-s + (−0.452 − 0.892i)8-s − 0.990i·9-s + (−0.0470 + 0.444i)10-s + (−1.05 + 1.05i)11-s + (−0.0822 + 0.0532i)12-s + (0.815 + 0.815i)13-s + (0.634 + 0.784i)14-s + 0.0438·15-s + (−0.912 − 0.409i)16-s + 0.448·17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.686+0.727i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.686+0.727i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.686+0.727i0.686 + 0.727i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(61,⋅)\chi_{80} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.686+0.727i)(2,\ 80,\ (\ :1/2),\ 0.686 + 0.727i)

Particular Values

L(1)L(1) ≈\approx 1.15620−0.498769i1.15620 - 0.498769i
L(12)L(\frac12) ≈\approx 1.15620−0.498769i1.15620 - 0.498769i
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+(−1.09+0.889i)T 1 + (-1.09 + 0.889i)T
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1+(0.120+0.120i)T+3iT2 1 + (0.120 + 0.120i)T + 3iT^{2}
7 1−2.66iT−7T2 1 - 2.66iT - 7T^{2}
11 1+(3.49−3.49i)T−11iT2 1 + (3.49 - 3.49i)T - 11iT^{2}
13 1+(−2.94−2.94i)T+13iT2 1 + (-2.94 - 2.94i)T + 13iT^{2}
17 1−1.85T+17T2 1 - 1.85T + 17T^{2}
19 1+(3.44+3.44i)T+19iT2 1 + (3.44 + 3.44i)T + 19iT^{2}
23 1+0.707iT−23T2 1 + 0.707iT - 23T^{2}
29 1+(3.49+3.49i)T+29iT2 1 + (3.49 + 3.49i)T + 29iT^{2}
31 1−6.84T+31T2 1 - 6.84T + 31T^{2}
37 1+(0.0975−0.0975i)T−37iT2 1 + (0.0975 - 0.0975i)T - 37iT^{2}
41 1+10.2iT−41T2 1 + 10.2iT - 41T^{2}
43 1+(−4.43+4.43i)T−43iT2 1 + (-4.43 + 4.43i)T - 43iT^{2}
47 1+1.89T+47T2 1 + 1.89T + 47T^{2}
53 1+(7.43−7.43i)T−53iT2 1 + (7.43 - 7.43i)T - 53iT^{2}
59 1+(−0.959+0.959i)T−59iT2 1 + (-0.959 + 0.959i)T - 59iT^{2}
61 1+(−6.49−6.49i)T+61iT2 1 + (-6.49 - 6.49i)T + 61iT^{2}
67 1+(−3.49−3.49i)T+67iT2 1 + (-3.49 - 3.49i)T + 67iT^{2}
71 1−7.86iT−71T2 1 - 7.86iT - 71T^{2}
73 1−15.6iT−73T2 1 - 15.6iT - 73T^{2}
79 1+6.70T+79T2 1 + 6.70T + 79T^{2}
83 1+(3.87+3.87i)T+83iT2 1 + (3.87 + 3.87i)T + 83iT^{2}
89 1−10.5iT−89T2 1 - 10.5iT - 89T^{2}
97 1−4.79T+97T2 1 - 4.79T + 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

−14.23999436590030247274426277155, −12.92967874858091376482797568087, −12.16718431993555287663477665261, −11.28318028511447054200071947552, −10.02731191576842261792548863641, −8.847625755503298800933982657784, −6.92545453694030598168826981481, −5.72702093116153551512454909827, −4.18279215762973442274404449718, −2.48720399837190436023573471264, 3.38028737164568903032151918570, 4.85044760492588689779600432482, 6.06093940610133685757631829097, 7.79989498170825628106330563327, 8.225908749823123915281804373468, 10.46440698273592368723416579831, 11.25937282014677062778248834614, 12.89971593051123853098059524432, 13.39572566766988987564733095866, 14.39847697888012679555592258747

Graph of the ZZ-function along the critical line