↑

Properties

Label 1-2e9-512.245-r0-0-0
Degree 11
Conductor 512512
Sign 0.953+0.302i0.953 + 0.302i
Analytic cond. 2.377712.37771
Root an. cond. 2.377712.37771
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0490 − 0.998i)3-s + (0.514 + 0.857i)5-s + (0.634 + 0.773i)7-s + (−0.995 + 0.0980i)9-s + (0.941 − 0.336i)11-s + (0.242 + 0.970i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (0.146 − 0.989i)19-s + (0.740 − 0.671i)21-s + (−0.290 + 0.956i)23-s + (−0.471 + 0.881i)25-s + (0.146 + 0.989i)27-s + (0.427 + 0.903i)29-s + (−0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (−0.0490 − 0.998i)3-s + (0.514 + 0.857i)5-s + (0.634 + 0.773i)7-s + (−0.995 + 0.0980i)9-s + (0.941 − 0.336i)11-s + (0.242 + 0.970i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (0.146 − 0.989i)19-s + (0.740 − 0.671i)21-s + (−0.290 + 0.956i)23-s + (−0.471 + 0.881i)25-s + (0.146 + 0.989i)27-s + (0.427 + 0.903i)29-s + (−0.382 + 0.923i)31-s + ⋯

Functional equation

Λ(s)=(512s/2 ΓR(s) L(s)=((0.953+0.302i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(512s/2 ΓR(s) L(s)=((0.953+0.302i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 512512    =    292^{9}
Sign: 0.953+0.302i0.953 + 0.302i
Analytic conductor: 2.377712.37771
Root analytic conductor: 2.377712.37771
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ512(245,⋅)\chi_{512} (245, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 512, (0: ), 0.953+0.302i)(1,\ 512,\ (0:\ ),\ 0.953 + 0.302i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.530829861+0.2366857605i1.530829861 + 0.2366857605i
L(12)L(\frac12) ≈\approx 1.530829861+0.2366857605i1.530829861 + 0.2366857605i
L(1)L(1) ≈\approx 1.218845819+0.005035299893i1.218845819 + 0.005035299893i
L(1)L(1) ≈\approx 1.218845819+0.005035299893i1.218845819 + 0.005035299893i

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
good3 1+(−0.0490−0.998i)T 1 + (-0.0490 - 0.998i)T
5 1+(0.514+0.857i)T 1 + (0.514 + 0.857i)T
7 1+(0.634+0.773i)T 1 + (0.634 + 0.773i)T
11 1+(0.941−0.336i)T 1 + (0.941 - 0.336i)T
13 1+(0.242+0.970i)T 1 + (0.242 + 0.970i)T
17 1+(−0.831−0.555i)T 1 + (-0.831 - 0.555i)T
19 1+(0.146−0.989i)T 1 + (0.146 - 0.989i)T
23 1+(−0.290+0.956i)T 1 + (-0.290 + 0.956i)T
29 1+(0.427+0.903i)T 1 + (0.427 + 0.903i)T
31 1+(−0.382+0.923i)T 1 + (-0.382 + 0.923i)T
37 1+(0.803+0.595i)T 1 + (0.803 + 0.595i)T
41 1+(−0.471−0.881i)T 1 + (-0.471 - 0.881i)T
43 1+(0.998+0.0490i)T 1 + (0.998 + 0.0490i)T
47 1+(0.195+0.980i)T 1 + (0.195 + 0.980i)T
53 1+(0.427−0.903i)T 1 + (0.427 - 0.903i)T
59 1+(−0.242+0.970i)T 1 + (-0.242 + 0.970i)T
61 1+(−0.740−0.671i)T 1 + (-0.740 - 0.671i)T
67 1+(0.671−0.740i)T 1 + (0.671 - 0.740i)T
71 1+(0.0980−0.995i)T 1 + (0.0980 - 0.995i)T
73 1+(0.634−0.773i)T 1 + (0.634 - 0.773i)T
79 1+(−0.980−0.195i)T 1 + (-0.980 - 0.195i)T
83 1+(0.803−0.595i)T 1 + (0.803 - 0.595i)T
89 1+(−0.290−0.956i)T 1 + (-0.290 - 0.956i)T
97 1+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
show more
show less
   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−23.400060768901990847914242631136, −22.629471432043311488705468427634, −21.78363597289115771593291666145, −20.92516027972586039061982698262, −20.2198723324637560993018779716, −19.89868259040728190572909719471, −18.19278626522944231668752868722, −17.21466230185933361766022929615, −16.951817495782158936364231496872, −15.96408214114949230013754065496, −14.94188965154665752692704714619, −14.23897172266301992305461631469, −13.29238942899257562179307841560, −12.25667946156247018307894599656, −11.228123139975478206616664837764, −10.33239020559081348223531983323, −9.65070106992332957304313285183, −8.62151620533134204772875191064, −7.931021629750403635640637670706, −6.32085323878049252279779442390, −5.49468265018190092683157301734, −4.332806081156152388540296837815, −3.97845667555580937062240757306, −2.270579838064357598817104524082, −0.92723806276324261295685379671, 1.439749116307085072511916291220, 2.244248272348536917661864665919, 3.264607724906605612426606248653, 4.86352463805299283634888286453, 6.00245319961642583326545623140, 6.6881794698920958107892932437, 7.45429948201600306964090752424, 8.825473819571685455538993550015, 9.254796835510365924065295365615, 10.980322613423635114501607887869, 11.441840025918725379257659726424, 12.24961207826437262514021878096, 13.56492499903197363372162752997, 14.032854157724690267472272301595, 14.80525777408425338632790944229, 15.91367095381508676309297805097, 17.20135050582826908948903436348, 17.85594156777888818884957003582, 18.43589382398637409085495844289, 19.27139068990553341829736336613, 20.00154370561318351448894369156, 21.37971552705614259102134916600, 21.95702339550362205653638439958, 22.69978114095606860568299827189, 23.92117699815791440211154893119

Graph of the ZZ-function along the critical line