↑

Properties

Label 1-1053-1053.254-r0-0-0
Degree 11
Conductor 10531053
Sign 0.941−0.336i0.941 - 0.336i
Analytic cond. 4.890114.89011
Root an. cond. 4.890114.89011
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  + (0.549 + 0.835i)2-s + (−0.396 + 0.918i)4-s + (0.230 − 0.973i)5-s + (0.918 − 0.396i)7-s + (−0.984 + 0.173i)8-s + (0.939 − 0.342i)10-s + (−0.957 − 0.286i)11-s + (0.835 + 0.549i)14-s + (−0.686 − 0.727i)16-s + (0.173 − 0.984i)17-s + (0.642 + 0.766i)19-s + (0.802 + 0.597i)20-s + (−0.286 − 0.957i)22-s + (0.396 − 0.918i)23-s + (−0.893 − 0.448i)25-s + ⋯
L(s)  = 1  + (0.549 + 0.835i)2-s + (−0.396 + 0.918i)4-s + (0.230 − 0.973i)5-s + (0.918 − 0.396i)7-s + (−0.984 + 0.173i)8-s + (0.939 − 0.342i)10-s + (−0.957 − 0.286i)11-s + (0.835 + 0.549i)14-s + (−0.686 − 0.727i)16-s + (0.173 − 0.984i)17-s + (0.642 + 0.766i)19-s + (0.802 + 0.597i)20-s + (−0.286 − 0.957i)22-s + (0.396 − 0.918i)23-s + (−0.893 − 0.448i)25-s + ⋯

Functional equation

Λ(s)=(1053s/2 ΓR(s) L(s)=((0.941−0.336i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1053s/2 ΓR(s) L(s)=((0.941−0.336i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10531053    =    34⋅133^{4} \cdot 13
Sign: 0.941−0.336i0.941 - 0.336i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(254,⋅)\chi_{1053} (254, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), 0.941−0.336i)(1,\ 1053,\ (0:\ ),\ 0.941 - 0.336i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.817023816−0.3144007408i1.817023816 - 0.3144007408i
L(12)L(\frac12) ≈\approx 1.817023816−0.3144007408i1.817023816 - 0.3144007408i
L(1)L(1) ≈\approx 1.365051569+0.2024970932i1.365051569 + 0.2024970932i
L(1)L(1) ≈\approx 1.365051569+0.2024970932i1.365051569 + 0.2024970932i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1 1
good2 1+(0.549+0.835i)T 1 + (0.549 + 0.835i)T
5 1+(0.230−0.973i)T 1 + (0.230 - 0.973i)T
7 1+(0.918−0.396i)T 1 + (0.918 - 0.396i)T
11 1+(−0.957−0.286i)T 1 + (-0.957 - 0.286i)T
17 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
19 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
23 1+(0.396−0.918i)T 1 + (0.396 - 0.918i)T
29 1+(−0.893−0.448i)T 1 + (-0.893 - 0.448i)T
31 1+(0.918+0.396i)T 1 + (0.918 + 0.396i)T
37 1+(−0.984−0.173i)T 1 + (-0.984 - 0.173i)T
41 1+(−0.448−0.893i)T 1 + (-0.448 - 0.893i)T
43 1+(0.686+0.727i)T 1 + (0.686 + 0.727i)T
47 1+(−0.918+0.396i)T 1 + (-0.918 + 0.396i)T
53 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
59 1+(0.957−0.286i)T 1 + (0.957 - 0.286i)T
61 1+(0.597−0.802i)T 1 + (0.597 - 0.802i)T
67 1+(−0.998−0.0581i)T 1 + (-0.998 - 0.0581i)T
71 1+(0.642−0.766i)T 1 + (0.642 - 0.766i)T
73 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
79 1+(−0.0581−0.998i)T 1 + (-0.0581 - 0.998i)T
83 1+(−0.998+0.0581i)T 1 + (-0.998 + 0.0581i)T
89 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
97 1+(−0.957−0.286i)T 1 + (-0.957 - 0.286i)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

−21.38313896675835988796564997511, −21.11154548829858491328146928219, −20.104497496110893474200914069037, −19.22237062528493282025575061104, −18.52559066779554460103758038338, −17.93172366942419297399074060274, −17.25143215217123843407753767744, −15.54110989478388949904898810062, −15.19476218524685862448390974697, −14.429499866282489779792622156282, −13.60285407508977411761020203125, −12.96041515137011584448822009467, −11.840923043071885087262889827472, −11.277275212116937005869715072947, −10.52929954037596072010068019900, −9.87888622520041986684189575316, −8.83555645843387156339520268922, −7.78320307208126147608441026199, −6.82657210591921510295316410856, −5.63362583409676924257292118845, −5.18981274655530821249648348757, −4.02599607488342944413468068111, −3.03082166873128360795023574852, −2.28566646039050097527161753740, −1.39893673031553747860494253212, 0.65134167692708109641222736623, 2.08985910969991543609676314794, 3.32176158970261588496679688525, 4.425582171666970506060835134962, 5.12009210399666630649023208351, 5.60163976122166015227337506786, 6.84826590911257808423744114141, 7.83294296689704830500171282606, 8.251390875231588728144720493570, 9.18247743931387589895234583267, 10.18840194727220528087748905994, 11.37422831525479675760804629298, 12.16115596765537650344209484196, 12.959923559653079935352372627473, 13.751980852191997604475873875819, 14.23936572455387201605356069907, 15.26060140593619767348911591176, 16.14530978620839899958989293311, 16.55714201001315923542520197607, 17.49940121145126855131111951028, 18.03708393442989287992542958851, 18.96640186377930219953829277350, 20.445733169707989846936409734059, 20.86126461300931164894360575586, 21.22996606435128188594652609099

Graph of the ZZ-function along the critical line