Properties

Label 1-800-800.779-r1-0-0
Degree 11
Conductor 800800
Sign 0.943+0.331i-0.943 + 0.331i
Analytic cond. 85.971985.9719
Root an. cond. 85.971985.9719
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.891 + 0.453i)3-s i·7-s + (0.587 + 0.809i)9-s + (0.156 + 0.987i)11-s + (−0.156 + 0.987i)13-s + (0.309 − 0.951i)17-s + (−0.453 − 0.891i)19-s + (−0.453 + 0.891i)21-s + (−0.587 + 0.809i)23-s + (0.156 + 0.987i)27-s + (0.891 + 0.453i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.987 − 0.156i)37-s + (−0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)3-s i·7-s + (0.587 + 0.809i)9-s + (0.156 + 0.987i)11-s + (−0.156 + 0.987i)13-s + (0.309 − 0.951i)17-s + (−0.453 − 0.891i)19-s + (−0.453 + 0.891i)21-s + (−0.587 + 0.809i)23-s + (0.156 + 0.987i)27-s + (0.891 + 0.453i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.987 − 0.156i)37-s + (−0.587 + 0.809i)39-s + ⋯

Functional equation

Λ(s)=(800s/2ΓR(s+1)L(s)=((0.943+0.331i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(800s/2ΓR(s+1)L(s)=((0.943+0.331i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 800800    =    25522^{5} \cdot 5^{2}
Sign: 0.943+0.331i-0.943 + 0.331i
Analytic conductor: 85.971985.9719
Root analytic conductor: 85.971985.9719
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ800(779,)\chi_{800} (779, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 800, (1: ), 0.943+0.331i)(1,\ 800,\ (1:\ ),\ -0.943 + 0.331i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.3799684810+2.228762210i0.3799684810 + 2.228762210i
L(12)L(\frac12) \approx 0.3799684810+2.228762210i0.3799684810 + 2.228762210i
L(1)L(1) \approx 1.196461047+0.6592294560i1.196461047 + 0.6592294560i
L(1)L(1) \approx 1.196461047+0.6592294560i1.196461047 + 0.6592294560i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+(0.891+0.453i)T 1 + (0.891 + 0.453i)T
7 1iT 1 - iT
11 1+(0.156+0.987i)T 1 + (0.156 + 0.987i)T
13 1+(0.156+0.987i)T 1 + (-0.156 + 0.987i)T
17 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
19 1+(0.4530.891i)T 1 + (-0.453 - 0.891i)T
23 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
29 1+(0.891+0.453i)T 1 + (0.891 + 0.453i)T
31 1+(0.309+0.951i)T 1 + (-0.309 + 0.951i)T
37 1+(0.9870.156i)T 1 + (-0.987 - 0.156i)T
41 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
43 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
47 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
53 1+(0.4530.891i)T 1 + (0.453 - 0.891i)T
59 1+(0.9870.156i)T 1 + (-0.987 - 0.156i)T
61 1+(0.987+0.156i)T 1 + (-0.987 + 0.156i)T
67 1+(0.4530.891i)T 1 + (-0.453 - 0.891i)T
71 1+(0.9510.309i)T 1 + (0.951 - 0.309i)T
73 1+(0.5870.809i)T 1 + (0.587 - 0.809i)T
79 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
83 1+(0.453+0.891i)T 1 + (0.453 + 0.891i)T
89 1+(0.5870.809i)T 1 + (0.587 - 0.809i)T
97 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
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   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

−21.52415827683957179623689671079, −20.71925447942360211003789091784, −20.14070271173987503190620891244, −19.2610069813026992380304570104, −18.7961662613985079402204772585, −17.65832384580732881298841722372, −16.97891919179595103576709684589, −16.01264284359210740851456392971, −15.05231846477880587601552086637, −14.22305089491121112498122482739, −13.71013920680552180006341135196, −12.78405738081055304724578514394, −12.14293922964949666068987666454, −10.66849223795124802528454527360, −10.28172664945795097654179500320, −9.083174566574143960172080792, −8.08174146923236115223995465453, −7.77605869816092889933442354531, −6.53896830371780428935490681749, −5.8014043023100791503109409471, −4.20004999552268975202379051391, −3.607975893302877523308788451271, −2.58204867061173302166492394753, −1.3611994440470848792626405374, −0.42134481896145305080555257678, 1.65505481805767116929737565904, 2.423003100294004604873275633144, 3.36117296601461950655829751250, 4.572788022242712626678944632489, 5.11968312433681392095443829216, 6.57875136282770770860188561664, 7.38520158756651284698645173408, 8.42833581429427614159540274038, 9.29278270995359054180895324309, 9.6296502844109400523244297085, 10.824369800296612516002794033107, 11.88780832805376205361966851350, 12.55251332416426536328641385441, 13.69547274645564004246317714225, 14.34624878208430235245343384996, 15.169905348608206259514875936120, 15.78667459868389767169638410796, 16.5485410828388972619923732247, 17.81910297035260688767373130155, 18.40458260069302661549504401722, 19.61779318667383235266794334844, 19.71455907212950007280737054880, 21.08644262796859199261931893776, 21.39103082900966791143589859062, 22.22801227467772730859461231391

Graph of the ZZ-function along the critical line