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Properties

Label 1-253-253.153-r0-0-0
Degree 11
Conductor 253253
Sign 0.294+0.955i0.294 + 0.955i
Analytic cond. 1.174921.17492
Root an. cond. 1.174921.17492
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.415 + 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.142 + 0.989i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.415 + 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.142 + 0.989i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓR(s) L(s)=((0.294+0.955i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(253s/2 ΓR(s) L(s)=((0.294+0.955i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.294+0.955i0.294 + 0.955i
Analytic conductor: 1.174921.17492
Root analytic conductor: 1.174921.17492
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ253(153,⋅)\chi_{253} (153, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 253, (0: ), 0.294+0.955i)(1,\ 253,\ (0:\ ),\ 0.294 + 0.955i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.518565434+1.120628394i1.518565434 + 1.120628394i
L(12)L(\frac12) ≈\approx 1.518565434+1.120628394i1.518565434 + 1.120628394i
L(1)L(1) ≈\approx 1.467413110+0.6354064129i1.467413110 + 0.6354064129i
L(1)L(1) ≈\approx 1.467413110+0.6354064129i1.467413110 + 0.6354064129i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad11 1 1
23 1 1
good2 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
3 1+(−0.654+0.755i)T 1 + (-0.654 + 0.755i)T
5 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
7 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
13 1+(−0.415+0.909i)T 1 + (-0.415 + 0.909i)T
17 1+(0.841−0.540i)T 1 + (0.841 - 0.540i)T
19 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)T
29 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
31 1+(−0.654−0.755i)T 1 + (-0.654 - 0.755i)T
37 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
41 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
43 1+(−0.654+0.755i)T 1 + (-0.654 + 0.755i)T
47 1+T 1 + T
53 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
59 1+(0.415−0.909i)T 1 + (0.415 - 0.909i)T
61 1+(−0.654−0.755i)T 1 + (-0.654 - 0.755i)T
67 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
71 1+(−0.959−0.281i)T 1 + (-0.959 - 0.281i)T
73 1+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
79 1+(0.415−0.909i)T 1 + (0.415 - 0.909i)T
83 1+(−0.142−0.989i)T 1 + (-0.142 - 0.989i)T
89 1+(0.654−0.755i)T 1 + (0.654 - 0.755i)T
97 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
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   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

−25.49654662305834166872256094247, −24.647589440914537134947581873893, −23.671743856484341925956053332398, −23.086488140850560861724509792763, −22.32475335122147444981766446608, −21.53271133853169758057520371565, −20.240387250216910251134372954925, −19.46433785491248547631374160719, −18.42531073696500477697988862107, −17.50682075842064125659090878743, −16.51822866530151385716250182657, −15.19181922025483584336198474925, −14.27053308525521776643571954155, −13.53520568680899064543073990377, −12.591994709077051378374734966718, −11.536547143937781816936630816271, −10.76433541757193867143260898562, −10.08242965383493233408392805596, −7.61039204066175611186117596429, −7.22785704655945458470403187567, −6.00177761899474598818851178145, −5.17744509894607981064640504726, −3.72421989073810963797049590938, −2.53853954883177238440691759310, −1.20830485021642298930303671689, 1.76977074649753253836895256235, 3.41909327144299864150473518598, 4.6494651542419042042281027708, 5.29012581478281540528782650343, 6.05464972217885321236075290809, 7.55706604896969231257376224938, 8.88195383875025592205994531565, 9.82262727674900022444257658480, 11.42500143859673835346879508745, 11.92107197507140494561135078595, 12.740588788012729447877754691434, 14.11533933806990876138124925767, 14.94265084143991595478279636333, 16.00954391189289529885125902080, 16.535585641169397446682992374335, 17.39019964824289661783429913107, 18.69084970572887079280497810742, 20.37899592098850025298873923486, 20.83949917509579042098174765772, 21.74915557526499918351580268042, 22.3169692709294928079552245071, 23.51461239881694044229587839399, 24.18197368033674218360378232873, 24.99950575434004372932874087951, 25.99074758743120425264387832732

Graph of the ZZ-function along the critical line