Properties

Label 1-405-405.148-r1-0-0
Degree 11
Conductor 405405
Sign 0.490+0.871i-0.490 + 0.871i
Analytic cond. 43.523243.5232
Root an. cond. 43.523243.5232
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  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.835 − 0.549i)11-s + (0.116 − 0.993i)13-s + (−0.597 + 0.802i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.549 + 0.835i)22-s + (0.230 + 0.973i)23-s + (−0.5 + 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.597 − 0.802i)29-s + ⋯
L(s)  = 1  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.835 − 0.549i)11-s + (0.116 − 0.993i)13-s + (−0.597 + 0.802i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (−0.766 − 0.642i)19-s + (0.549 + 0.835i)22-s + (0.230 + 0.973i)23-s + (−0.5 + 0.866i)26-s + (0.866 − 0.5i)28-s + (−0.597 − 0.802i)29-s + ⋯

Functional equation

Λ(s)=(405s/2ΓR(s+1)L(s)=((0.490+0.871i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(405s/2ΓR(s+1)L(s)=((0.490+0.871i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 405405    =    3453^{4} \cdot 5
Sign: 0.490+0.871i-0.490 + 0.871i
Analytic conductor: 43.523243.5232
Root analytic conductor: 43.523243.5232
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ405(148,)\chi_{405} (148, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 405, (1: ), 0.490+0.871i)(1,\ 405,\ (1:\ ),\ -0.490 + 0.871i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.069305222450.1185865562i-0.06930522245 - 0.1185865562i
L(12)L(\frac12) \approx 0.069305222450.1185865562i-0.06930522245 - 0.1185865562i
L(1)L(1) \approx 0.53598061330.2243017398i0.5359806133 - 0.2243017398i
L(1)L(1) \approx 0.53598061330.2243017398i0.5359806133 - 0.2243017398i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
good2 1+(0.9180.396i)T 1 + (-0.918 - 0.396i)T
7 1+(0.2300.973i)T 1 + (0.230 - 0.973i)T
11 1+(0.8350.549i)T 1 + (-0.835 - 0.549i)T
13 1+(0.1160.993i)T 1 + (0.116 - 0.993i)T
17 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
19 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
23 1+(0.230+0.973i)T 1 + (0.230 + 0.973i)T
29 1+(0.5970.802i)T 1 + (-0.597 - 0.802i)T
31 1+(0.286+0.957i)T 1 + (-0.286 + 0.957i)T
37 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
41 1+(0.396+0.918i)T 1 + (0.396 + 0.918i)T
43 1+(0.448+0.893i)T 1 + (0.448 + 0.893i)T
47 1+(0.957+0.286i)T 1 + (-0.957 + 0.286i)T
53 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
59 1+(0.8350.549i)T 1 + (0.835 - 0.549i)T
61 1+(0.686+0.727i)T 1 + (-0.686 + 0.727i)T
67 1+(0.8020.597i)T 1 + (-0.802 - 0.597i)T
71 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
73 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
79 1+(0.396+0.918i)T 1 + (-0.396 + 0.918i)T
83 1+(0.918+0.396i)T 1 + (0.918 + 0.396i)T
89 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
97 1+(0.998+0.0581i)T 1 + (0.998 + 0.0581i)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

−24.734830762352884407790830771151, −24.06351209975682946120103581113, −23.26666730334326776094572525988, −22.016153117910800065626966371256, −21.05228239507996475639322470333, −20.33866395498767985026021497758, −19.10204692259449993758505171614, −18.61948310371140654024392855156, −17.82451129683088305592071930104, −16.832862019712032792786530616882, −16.04015569045251653751388383236, −15.03352189102122191223591332747, −14.61059662574857998117779404046, −13.06814618993929078699251014259, −12.07936995436472836587430917942, −11.06031856618238038802857491192, −10.25194230076791623790352152382, −9.09561481829097412774649383698, −8.52096230977486058545475487877, −7.47720177346305807622267398758, −6.40715254235213466963011272394, −5.56205761193936535053268910156, −4.34474655301293708461074841341, −2.45322245845773125411715433728, −1.79111697659162914630577337601, 0.05596108957956790466864693234, 1.03136206126137478789981356739, 2.49611072505088546353564100728, 3.46488694007325160437877132443, 4.775348640252864484865426174432, 6.20843332856469335694456019570, 7.40097030656535824615365113564, 7.97497459612644936265762833585, 9.06692451593704286956053449427, 10.09942836853401088920666462997, 10.91376820002951390687602455310, 11.46538238839206640932826446687, 13.01713502892237984043956695031, 13.36712674152527561576033092587, 14.91877410541089758121604986252, 15.88588061871991013082028452025, 16.62788835885167413309963579106, 17.73385844121430591472462234957, 18.041102695151963679803583813548, 19.37362876835298105000022838967, 19.90902276844050445890780135435, 20.851112675470122767373845323508, 21.45141170146603046205072872120, 22.66440396088730810208619234346, 23.62235108137563508782640360764

Graph of the ZZ-function along the critical line