↑

Properties

Label 1-1037-1037.100-r0-0-0
Degree 11
Conductor 10371037
Sign 0.869−0.493i0.869 - 0.493i
Analytic cond. 4.815804.81580
Root an. cond. 4.815804.81580
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  + (0.994 + 0.104i)2-s + (0.987 − 0.156i)3-s + (0.978 + 0.207i)4-s + (−0.0523 − 0.998i)5-s + (0.998 − 0.0523i)6-s + (−0.358 − 0.933i)7-s + (0.951 + 0.309i)8-s + (0.951 − 0.309i)9-s + (0.0523 − 0.998i)10-s + (0.707 + 0.707i)11-s + (0.998 + 0.0523i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.207 − 0.978i)15-s + (0.913 + 0.406i)16-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.987 − 0.156i)3-s + (0.978 + 0.207i)4-s + (−0.0523 − 0.998i)5-s + (0.998 − 0.0523i)6-s + (−0.358 − 0.933i)7-s + (0.951 + 0.309i)8-s + (0.951 − 0.309i)9-s + (0.0523 − 0.998i)10-s + (0.707 + 0.707i)11-s + (0.998 + 0.0523i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.207 − 0.978i)15-s + (0.913 + 0.406i)16-s + ⋯

Functional equation

Λ(s)=(1037s/2 ΓR(s) L(s)=((0.869−0.493i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1037s/2 ΓR(s) L(s)=((0.869−0.493i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10371037    =    17⋅6117 \cdot 61
Sign: 0.869−0.493i0.869 - 0.493i
Analytic conductor: 4.815804.81580
Root analytic conductor: 4.815804.81580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1037(100,⋅)\chi_{1037} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1037, (0: ), 0.869−0.493i)(1,\ 1037,\ (0:\ ),\ 0.869 - 0.493i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 4.211046251−1.112345943i4.211046251 - 1.112345943i
L(12)L(\frac12) ≈\approx 4.211046251−1.112345943i4.211046251 - 1.112345943i
L(1)L(1) ≈\approx 2.684664380−0.4081264189i2.684664380 - 0.4081264189i
L(1)L(1) ≈\approx 2.684664380−0.4081264189i2.684664380 - 0.4081264189i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad17 1 1
61 1 1
good2 1+(0.994+0.104i)T 1 + (0.994 + 0.104i)T
3 1+(0.987−0.156i)T 1 + (0.987 - 0.156i)T
5 1+(−0.0523−0.998i)T 1 + (-0.0523 - 0.998i)T
7 1+(−0.358−0.933i)T 1 + (-0.358 - 0.933i)T
11 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
13 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(0.406+0.913i)T 1 + (0.406 + 0.913i)T
23 1+(−0.453+0.891i)T 1 + (-0.453 + 0.891i)T
29 1+(−0.258−0.965i)T 1 + (-0.258 - 0.965i)T
31 1+(−0.777−0.629i)T 1 + (-0.777 - 0.629i)T
37 1+(−0.156+0.987i)T 1 + (-0.156 + 0.987i)T
41 1+(−0.987−0.156i)T 1 + (-0.987 - 0.156i)T
43 1+(−0.207−0.978i)T 1 + (-0.207 - 0.978i)T
47 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
53 1+(−0.951−0.309i)T 1 + (-0.951 - 0.309i)T
59 1+(−0.994−0.104i)T 1 + (-0.994 - 0.104i)T
67 1+(−0.669+0.743i)T 1 + (-0.669 + 0.743i)T
71 1+(0.998+0.0523i)T 1 + (0.998 + 0.0523i)T
73 1+(0.998+0.0523i)T 1 + (0.998 + 0.0523i)T
79 1+(−0.838−0.544i)T 1 + (-0.838 - 0.544i)T
83 1+(−0.994−0.104i)T 1 + (-0.994 - 0.104i)T
89 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
97 1+(−0.777−0.629i)T 1 + (-0.777 - 0.629i)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

−21.90959876132909532812793399301, −21.02301058169935839432091521548, −19.95234245674299898798569743412, −19.62052590861853482065207422976, −18.65593757364158884861712644301, −18.0993953871778037637155972896, −16.5068677289241970863053527278, −15.684237746659063551506897044196, −15.27343503864877272656515083190, −14.36232432701685084124425272201, −14.01713399249355288734594410801, −12.984716418584544550742915539856, −12.34785031620215144116073397568, −11.1964400050624960520882739455, −10.666609496239164516029834369894, −9.60131476866941142075583178409, −8.708369346164835422091157380143, −7.72520561583038176340756106311, −6.77202282134234909897343049297, −6.10106622674809396594001042980, −5.09484938429252703995554421888, −3.84918124586051220117599779067, −3.13811371425287254778170442341, −2.707274691160002043422966810228, −1.57310976896554408883513519649, 1.429721778698226778575639467760, 1.90039383168130536843122651709, 3.52022329646793418010953286023, 3.93185379921178368516772717513, 4.64569266609762724193372390300, 5.896756366185265204675801926015, 6.88798011302148625291411283643, 7.559464082552552962252543710618, 8.40033960208234076471517499623, 9.50360164205269218173075767063, 10.08027434487008846502920141756, 11.51441900906897845827805511521, 12.1838623215503243292193701034, 13.00798029015946050629248247667, 13.72106756330885375981194652715, 14.07103232050053585985354269664, 15.15590999952035578377858559963, 15.81878184032830438186251701360, 16.70738833189971305767519633154, 17.18322269758048911504158094473, 18.67900826596109194522287267035, 19.554774793594080713029017908978, 20.29408539489702957219757171292, 20.5017051509122962507554225445, 21.34202822980670133171184751053

Graph of the ZZ-function along the critical line