Properties

Label 1-1053-1053.1019-r0-0-0
Degree 11
Conductor 10531053
Sign 0.9000.434i-0.900 - 0.434i
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  + (0.727 − 0.686i)2-s + (0.0581 − 0.998i)4-s + (−0.116 − 0.993i)5-s + (−0.448 + 0.893i)7-s + (−0.642 − 0.766i)8-s + (−0.766 − 0.642i)10-s + (0.918 − 0.396i)11-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.984 − 0.173i)19-s + (−0.998 + 0.0581i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + (−0.973 + 0.230i)25-s + ⋯
L(s)  = 1  + (0.727 − 0.686i)2-s + (0.0581 − 0.998i)4-s + (−0.116 − 0.993i)5-s + (−0.448 + 0.893i)7-s + (−0.642 − 0.766i)8-s + (−0.766 − 0.642i)10-s + (0.918 − 0.396i)11-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.984 − 0.173i)19-s + (−0.998 + 0.0581i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + (−0.973 + 0.230i)25-s + ⋯

Functional equation

Λ(s)=(1053s/2ΓR(s)L(s)=((0.9000.434i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1053s/2ΓR(s)L(s)=((0.9000.434i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10531053    =    34133^{4} \cdot 13
Sign: 0.9000.434i-0.900 - 0.434i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(1019,)\chi_{1053} (1019, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), 0.9000.434i)(1,\ 1053,\ (0:\ ),\ -0.900 - 0.434i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.43197990191.891133727i0.4319799019 - 1.891133727i
L(12)L(\frac12) \approx 0.43197990191.891133727i0.4319799019 - 1.891133727i
L(1)L(1) \approx 1.0941123350.9487987528i1.094112335 - 0.9487987528i
L(1)L(1) \approx 1.0941123350.9487987528i1.094112335 - 0.9487987528i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1 1
good2 1+(0.7270.686i)T 1 + (0.727 - 0.686i)T
5 1+(0.1160.993i)T 1 + (-0.116 - 0.993i)T
7 1+(0.448+0.893i)T 1 + (-0.448 + 0.893i)T
11 1+(0.9180.396i)T 1 + (0.918 - 0.396i)T
17 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
19 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
23 1+(0.8930.448i)T 1 + (0.893 - 0.448i)T
29 1+(0.2860.957i)T 1 + (0.286 - 0.957i)T
31 1+(0.549+0.835i)T 1 + (-0.549 + 0.835i)T
37 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
41 1+(0.7270.686i)T 1 + (-0.727 - 0.686i)T
43 1+(0.597+0.802i)T 1 + (-0.597 + 0.802i)T
47 1+(0.549+0.835i)T 1 + (0.549 + 0.835i)T
53 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
59 1+(0.9180.396i)T 1 + (-0.918 - 0.396i)T
61 1+(0.05810.998i)T 1 + (-0.0581 - 0.998i)T
67 1+(0.957+0.286i)T 1 + (-0.957 + 0.286i)T
71 1+(0.642+0.766i)T 1 + (-0.642 + 0.766i)T
73 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
79 1+(0.6860.727i)T 1 + (-0.686 - 0.727i)T
83 1+(0.7270.686i)T 1 + (0.727 - 0.686i)T
89 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
97 1+(0.116+0.993i)T 1 + (-0.116 + 0.993i)T
show more
show less
   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.25001982027779823426176889761, −21.34482042155717153602343795476, −20.270324631019267882878733257004, −19.69412934229826411586492024901, −18.647628616835440395798241396135, −17.7876512355595047867701967677, −16.91693653999165845860932514149, −16.51820371553995315924960069509, −15.12735020686072278827112914191, −15.05232557849157032783916061863, −13.896399498862589293257187689812, −13.54879975576886650477307091745, −12.42755013177650054339111648675, −11.684411952038979474592335283909, −10.77750263449023730541905579716, −9.90980095979584962826847592018, −8.874684953971608246048686473803, −7.69254458523652869152757061701, −7.08259922815284361274521459507, −6.51863966554143765980133746878, −5.595474625623098514506595351849, −4.40033872501894186267812856477, −3.584525777578290147464218146507, −3.062487644784988914223839766, −1.555568923955259663835138154429, 0.64915963266124498905062060557, 1.67055621773747460005069536738, 2.83739632136850266847396492050, 3.61031134603136025528302958162, 4.7240558457949577284193438401, 5.35719809650905593744060880464, 6.1697370621010229246693551074, 7.18945629089532990892879245674, 8.65584107191622019067746049291, 9.22161841709110356086138118522, 9.84614889908052677292169268590, 11.16786480128226626289199367643, 11.86164078885456512877210535334, 12.346538471342916442293723917081, 13.18425882270644783288454794812, 13.93737661172613242780217755383, 14.75129912278339934286712940136, 15.82450543772515118239819438349, 16.14287729969802925296160011721, 17.28650942958784013116011419228, 18.3444029073607962695355769205, 19.10190098046030407164258524844, 19.758626054910318347379831206763, 20.4713316922022079986854227078, 21.21133066053041501904114094057

Graph of the ZZ-function along the critical line