↑

Properties

Label 1-547-547.131-r0-0-0
Degree 11
Conductor 547547
Sign 0.0401+0.999i0.0401 + 0.999i
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  + (0.978 + 0.205i)2-s + (0.623 + 0.781i)3-s + (0.915 + 0.402i)4-s + (0.813 − 0.582i)5-s + (0.449 + 0.893i)6-s + (−0.792 + 0.609i)7-s + (0.813 + 0.582i)8-s + (−0.222 + 0.974i)9-s + (0.915 − 0.402i)10-s + (−0.354 + 0.935i)11-s + (0.256 + 0.966i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (0.962 + 0.272i)15-s + (0.675 + 0.736i)16-s + (0.990 − 0.137i)17-s + ⋯
L(s)  = 1  + (0.978 + 0.205i)2-s + (0.623 + 0.781i)3-s + (0.915 + 0.402i)4-s + (0.813 − 0.582i)5-s + (0.449 + 0.893i)6-s + (−0.792 + 0.609i)7-s + (0.813 + 0.582i)8-s + (−0.222 + 0.974i)9-s + (0.915 − 0.402i)10-s + (−0.354 + 0.935i)11-s + (0.256 + 0.966i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (0.962 + 0.272i)15-s + (0.675 + 0.736i)16-s + (0.990 − 0.137i)17-s + ⋯

Functional equation

Λ(s)=(547s/2 ΓR(s) L(s)=((0.0401+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0401 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(547s/2 ΓR(s) L(s)=((0.0401+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0401 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 547547
Sign: 0.0401+0.999i0.0401 + 0.999i
Analytic conductor: 2.540252.54025
Root analytic conductor: 2.540252.54025
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ547(131,⋅)\chi_{547} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 547, (0: ), 0.0401+0.999i)(1,\ 547,\ (0:\ ),\ 0.0401 + 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.293220930+2.202994194i2.293220930 + 2.202994194i
L(12)L(\frac12) ≈\approx 2.293220930+2.202994194i2.293220930 + 2.202994194i
L(1)L(1) ≈\approx 2.031703155+1.033986391i2.031703155 + 1.033986391i
L(1)L(1) ≈\approx 2.031703155+1.033986391i2.031703155 + 1.033986391i

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+(0.978+0.205i)T 1 + (0.978 + 0.205i)T
3 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
5 1+(0.813−0.582i)T 1 + (0.813 - 0.582i)T
7 1+(−0.792+0.609i)T 1 + (-0.792 + 0.609i)T
11 1+(−0.354+0.935i)T 1 + (-0.354 + 0.935i)T
13 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
17 1+(0.990−0.137i)T 1 + (0.990 - 0.137i)T
19 1+(−0.868−0.495i)T 1 + (-0.868 - 0.495i)T
23 1+(−0.539−0.842i)T 1 + (-0.539 - 0.842i)T
29 1+(−0.994+0.103i)T 1 + (-0.994 + 0.103i)T
31 1+(0.997−0.0689i)T 1 + (0.997 - 0.0689i)T
37 1+(0.940+0.338i)T 1 + (0.940 + 0.338i)T
41 1+T 1 + T
43 1+(0.188−0.982i)T 1 + (0.188 - 0.982i)T
47 1+(0.120−0.992i)T 1 + (0.120 - 0.992i)T
53 1+(0.962+0.272i)T 1 + (0.962 + 0.272i)T
59 1+(0.568−0.822i)T 1 + (0.568 - 0.822i)T
61 1+(−0.0172−0.999i)T 1 + (-0.0172 - 0.999i)T
67 1+(0.256+0.966i)T 1 + (0.256 + 0.966i)T
71 1+(−0.928+0.370i)T 1 + (-0.928 + 0.370i)T
73 1+(0.962−0.272i)T 1 + (0.962 - 0.272i)T
79 1+(0.188−0.982i)T 1 + (0.188 - 0.982i)T
83 1+(−0.748−0.663i)T 1 + (-0.748 - 0.663i)T
89 1+(−0.0172+0.999i)T 1 + (-0.0172 + 0.999i)T
97 1+(0.725+0.688i)T 1 + (0.725 + 0.688i)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

−23.097597205200112232751459815202, −22.489298252302483659975672849294, −21.364605603518779393288745028110, −20.93827827704752239986024408487, −19.64076856366144277625578723547, −19.330858682281662743820875463613, −18.40798928087324301318937397923, −17.21768777105049076757952385527, −16.37657425943818086102710156488, −15.1110555024935201695318745049, −14.40732862360812871984887482016, −13.73529214547499725380847272049, −13.05783150334410602411603077878, −12.42376755313896238250464609046, −11.2093330879078638615697273271, −10.18871775236918604560478549769, −9.55899131519347467405071397571, −7.89040139806581663697560944576, −7.2165746272020846251480441463, −6.11333762281037836011667811629, −5.737541921175722355501488984149, −3.97595173461688436627314721751, −3.05428264708708474722940988221, −2.445454330557510363520688080749, −1.14657206094188044800852600102, 2.21969939097643781473667475345, 2.53584883507772709547930076005, 3.95232282678318819744894065046, 4.82287225319105351298819969870, 5.540594919266159990011586678038, 6.596789168201003075254205620316, 7.75909921897572066629394070572, 8.86967126385448069939036979031, 9.79306741519842062585331344406, 10.37164002817329566232453665887, 11.8949691933320435408563664092, 12.68618646293660853366383125796, 13.32875563506017259429805142993, 14.388607666247435796737036797, 14.943562013462852535363158560760, 15.83974391325181791827851495471, 16.61400151132403803703560977325, 17.2421948748517012218385217943, 18.76839876769026639149889230736, 19.787498360871853612217461527540, 20.5031418685953342066298066792, 21.220198700912469092646680867423, 21.88557721411963011603908799255, 22.475898657144228956323201436244, 23.500902648729771971849612447630

Graph of the ZZ-function along the critical line