↑

Properties

Label 1-547-547.149-r0-0-0
Degree 11
Conductor 547547
Sign 0.278+0.960i0.278 + 0.960i
Analytic cond. 2.540252.54025
Root an. cond. 2.540252.54025
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  + (−0.700 + 0.713i)2-s + (−0.222 + 0.974i)3-s + (−0.0172 − 0.999i)4-s + (0.725 − 0.688i)5-s + (−0.539 − 0.842i)6-s + (−0.994 + 0.103i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (0.568 + 0.822i)11-s + (0.978 + 0.205i)12-s + (0.623 − 0.781i)13-s + (0.623 − 0.781i)14-s + (0.509 + 0.860i)15-s + (−0.999 + 0.0345i)16-s + (−0.868 + 0.495i)17-s + ⋯
L(s)  = 1  + (−0.700 + 0.713i)2-s + (−0.222 + 0.974i)3-s + (−0.0172 − 0.999i)4-s + (0.725 − 0.688i)5-s + (−0.539 − 0.842i)6-s + (−0.994 + 0.103i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (0.568 + 0.822i)11-s + (0.978 + 0.205i)12-s + (0.623 − 0.781i)13-s + (0.623 − 0.781i)14-s + (0.509 + 0.860i)15-s + (−0.999 + 0.0345i)16-s + (−0.868 + 0.495i)17-s + ⋯

Functional equation

Λ(s)=(547s/2 ΓR(s) L(s)=((0.278+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(547s/2 ΓR(s) L(s)=((0.278+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 547547
Sign: 0.278+0.960i0.278 + 0.960i
Analytic conductor: 2.540252.54025
Root analytic conductor: 2.540252.54025
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ547(149,⋅)\chi_{547} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 547, (0: ), 0.278+0.960i)(1,\ 547,\ (0:\ ),\ 0.278 + 0.960i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.7019757379+0.5272079803i0.7019757379 + 0.5272079803i
L(12)L(\frac12) ≈\approx 0.7019757379+0.5272079803i0.7019757379 + 0.5272079803i
L(1)L(1) ≈\approx 0.6736621583+0.3381314652i0.6736621583 + 0.3381314652i
L(1)L(1) ≈\approx 0.6736621583+0.3381314652i0.6736621583 + 0.3381314652i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad547 1 1
good2 1+(−0.700+0.713i)T 1 + (-0.700 + 0.713i)T
3 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
5 1+(0.725−0.688i)T 1 + (0.725 - 0.688i)T
7 1+(−0.994+0.103i)T 1 + (-0.994 + 0.103i)T
11 1+(0.568+0.822i)T 1 + (0.568 + 0.822i)T
13 1+(0.623−0.781i)T 1 + (0.623 - 0.781i)T
17 1+(−0.868+0.495i)T 1 + (-0.868 + 0.495i)T
19 1+(0.915−0.402i)T 1 + (0.915 - 0.402i)T
23 1+(−0.985+0.171i)T 1 + (-0.985 + 0.171i)T
29 1+(0.386−0.922i)T 1 + (0.386 - 0.922i)T
31 1+(0.256+0.966i)T 1 + (0.256 + 0.966i)T
37 1+(0.962−0.272i)T 1 + (0.962 - 0.272i)T
41 1+T 1 + T
43 1+(0.449+0.893i)T 1 + (0.449 + 0.893i)T
47 1+(−0.748+0.663i)T 1 + (-0.748 + 0.663i)T
53 1+(0.509+0.860i)T 1 + (0.509 + 0.860i)T
59 1+(0.885−0.464i)T 1 + (0.885 - 0.464i)T
61 1+(0.322−0.946i)T 1 + (0.322 - 0.946i)T
67 1+(0.978+0.205i)T 1 + (0.978 + 0.205i)T
71 1+(−0.596−0.802i)T 1 + (-0.596 - 0.802i)T
73 1+(0.509−0.860i)T 1 + (0.509 - 0.860i)T
79 1+(0.449+0.893i)T 1 + (0.449 + 0.893i)T
83 1+(−0.354+0.935i)T 1 + (-0.354 + 0.935i)T
89 1+(0.322+0.946i)T 1 + (0.322 + 0.946i)T
97 1+(−0.289−0.957i)T 1 + (-0.289 - 0.957i)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

−22.89188494567923692228229059658, −22.28805600025558250943804243070, −21.6949558820338203640421758694, −20.43928120121488395160188604535, −19.64474512632117904647524571737, −18.88027009469380173564030217073, −18.33947307740340424346250075332, −17.66240195391616301081952971927, −16.601732827604696248536866119798, −16.107783353330190822036072449971, −14.23108039638466120317679824179, −13.592090760550101061227542472512, −12.98518431169112699437147563133, −11.76551844057554605553619197120, −11.291274775335205291013584395537, −10.22122589803127631727479498977, −9.32776350598736998370195095052, −8.52739011718004524818097491717, −7.2467583210423090364414299438, −6.58698572717971182650876294438, −5.83587823429014021356098669128, −3.87586075694014182064499382169, −2.89058939687264595642368260152, −2.02890225666614924587763768570, −0.84619250662094945073179731297, 0.90885515384905693378636027847, 2.46235941222309513305170743480, 4.02243451264724335053683269564, 4.98873200448438133072654714301, 5.99577988831319018433183761213, 6.461740010443175665778210754390, 7.992987974163483888870978843768, 9.006515033217653736354900874238, 9.60135250245680932894982413041, 10.12040877539075568945091955488, 11.17149100536865367020307991388, 12.448209636347941786076765192818, 13.49677866248112583869861761992, 14.4097759244327406596419030235, 15.56729519789304502798164580334, 15.89934489612389682201742508600, 16.731461074065722732559033687158, 17.698418563098445252162763743, 17.92469163682739886458126923438, 19.72754760624619112088565748072, 19.927539129408367309651912279275, 20.94644245810657245427828642094, 22.1019361644971251041951058949, 22.664989718545977705470312088602, 23.52611922371065120428476468794

Graph of the ZZ-function along the critical line