↑

Properties

Label 1-1053-1053.2-r0-0-0
Degree 11
Conductor 10531053
Sign −0.736+0.676i-0.736 + 0.676i
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  + (0.802 + 0.597i)2-s + (0.286 + 0.957i)4-s + (0.448 + 0.893i)5-s + (0.230 + 0.973i)7-s + (−0.342 + 0.939i)8-s + (−0.173 + 0.984i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)14-s + (−0.835 + 0.549i)16-s + (0.173 − 0.984i)17-s + (−0.342 + 0.939i)19-s + (−0.727 + 0.686i)20-s + (0.893 − 0.448i)22-s + (0.973 + 0.230i)23-s + (−0.597 + 0.802i)25-s + ⋯
L(s)  = 1  + (0.802 + 0.597i)2-s + (0.286 + 0.957i)4-s + (0.448 + 0.893i)5-s + (0.230 + 0.973i)7-s + (−0.342 + 0.939i)8-s + (−0.173 + 0.984i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)14-s + (−0.835 + 0.549i)16-s + (0.173 − 0.984i)17-s + (−0.342 + 0.939i)19-s + (−0.727 + 0.686i)20-s + (0.893 − 0.448i)22-s + (0.973 + 0.230i)23-s + (−0.597 + 0.802i)25-s + ⋯

Functional equation

Λ(s)=(1053s/2 ΓR(s) L(s)=((−0.736+0.676i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1053s/2 ΓR(s) L(s)=((−0.736+0.676i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10531053    =    34⋅133^{4} \cdot 13
Sign: −0.736+0.676i-0.736 + 0.676i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(2,⋅)\chi_{1053} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), −0.736+0.676i)(1,\ 1053,\ (0:\ ),\ -0.736 + 0.676i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.9790659515+2.511901410i0.9790659515 + 2.511901410i
L(12)L(\frac12) ≈\approx 0.9790659515+2.511901410i0.9790659515 + 2.511901410i
L(1)L(1) ≈\approx 1.385138891+1.182399946i1.385138891 + 1.182399946i
L(1)L(1) ≈\approx 1.385138891+1.182399946i1.385138891 + 1.182399946i

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+(0.802+0.597i)T 1 + (0.802 + 0.597i)T
5 1+(0.448+0.893i)T 1 + (0.448 + 0.893i)T
7 1+(0.230+0.973i)T 1 + (0.230 + 0.973i)T
11 1+(0.448−0.893i)T 1 + (0.448 - 0.893i)T
17 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
19 1+(−0.342+0.939i)T 1 + (-0.342 + 0.939i)T
23 1+(0.973+0.230i)T 1 + (0.973 + 0.230i)T
29 1+(0.993+0.116i)T 1 + (0.993 + 0.116i)T
31 1+(0.727+0.686i)T 1 + (0.727 + 0.686i)T
37 1+(−0.642+0.766i)T 1 + (-0.642 + 0.766i)T
41 1+(0.918+0.396i)T 1 + (0.918 + 0.396i)T
43 1+(−0.893−0.448i)T 1 + (-0.893 - 0.448i)T
47 1+(−0.727+0.686i)T 1 + (-0.727 + 0.686i)T
53 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
59 1+(−0.448−0.893i)T 1 + (-0.448 - 0.893i)T
61 1+(−0.686−0.727i)T 1 + (-0.686 - 0.727i)T
67 1+(0.802−0.597i)T 1 + (0.802 - 0.597i)T
71 1+(−0.642+0.766i)T 1 + (-0.642 + 0.766i)T
73 1+(0.342−0.939i)T 1 + (0.342 - 0.939i)T
79 1+(−0.993−0.116i)T 1 + (-0.993 - 0.116i)T
83 1+(0.116−0.993i)T 1 + (0.116 - 0.993i)T
89 1+(−0.642−0.766i)T 1 + (-0.642 - 0.766i)T
97 1+(−0.998+0.0581i)T 1 + (-0.998 + 0.0581i)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

−21.15797321954953183803622072186, −20.60525207611455636034552325095, −19.68974387213581379083848694977, −19.484398819582723682941650458222, −18.00705133856539641756622221335, −17.27523925514427812120750020128, −16.64728784874085497260799030278, −15.51405005574973605631064780813, −14.80192251588081411110035465427, −13.930596947965371902451951758780, −13.24532988257995934160338177903, −12.63790155999632259366274278773, −11.865447899409406286453541870690, −10.85490295771922688620402435484, −10.168634155030225141756559688253, −9.39403292918626649883344189993, −8.43881886777758673114689070515, −7.16007739695325044749485639373, −6.41444290080486352319350798896, −5.35086774438372060662643738367, −4.46318699471230474258198378559, −4.0967157353681030794925785079, −2.70285352508473881318728403536, −1.6514821232340347034027790990, −0.887546274816796435285571007925, 1.71666003624758620610404992301, 2.94067481459664198358223634015, 3.25736101109060064961627061485, 4.7080567812914488529151565078, 5.50818497387619089087509834908, 6.298496472267303786632475276247, 6.86813154481890839768965379004, 8.02241892677912052230033431751, 8.7308693079846459572115101199, 9.74120781566093833722694973933, 10.936759188296409202267155565819, 11.62845888397070946184966918558, 12.32271244721817068283104274556, 13.381794435312755437058493601077, 14.145130235178116988539335613827, 14.58921770540827924226195132191, 15.461880073454907269500884071141, 16.14388286619786810400745380537, 17.07788015387117193051648108094, 17.84835515104551994641328906195, 18.6394279905738699463060271354, 19.30859563635234941444042478070, 20.64704651612040176928082825847, 21.49212928911119494748036118414, 21.66531067338668128570423475574

Graph of the ZZ-function along the critical line