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Properties

Label 1-19e2-361.72-r1-0-0
Degree 11
Conductor 361361
Sign 0.618−0.786i0.618 - 0.786i
Analytic cond. 38.794838.7948
Root an. cond. 38.794838.7948
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  + (0.435 − 0.900i)2-s + (0.263 + 0.964i)3-s + (−0.621 − 0.783i)4-s + (−0.729 − 0.684i)5-s + (0.983 + 0.182i)6-s + (0.0275 + 0.999i)7-s + (−0.975 + 0.218i)8-s + (−0.861 + 0.507i)9-s + (−0.933 + 0.359i)10-s + (0.350 − 0.936i)11-s + (0.592 − 0.805i)12-s + (−0.418 + 0.908i)13-s + (0.912 + 0.410i)14-s + (0.467 − 0.883i)15-s + (−0.227 + 0.973i)16-s + (−0.367 − 0.929i)17-s + ⋯
L(s)  = 1  + (0.435 − 0.900i)2-s + (0.263 + 0.964i)3-s + (−0.621 − 0.783i)4-s + (−0.729 − 0.684i)5-s + (0.983 + 0.182i)6-s + (0.0275 + 0.999i)7-s + (−0.975 + 0.218i)8-s + (−0.861 + 0.507i)9-s + (−0.933 + 0.359i)10-s + (0.350 − 0.936i)11-s + (0.592 − 0.805i)12-s + (−0.418 + 0.908i)13-s + (0.912 + 0.410i)14-s + (0.467 − 0.883i)15-s + (−0.227 + 0.973i)16-s + (−0.367 − 0.929i)17-s + ⋯

Functional equation

Λ(s)=(361s/2 ΓR(s+1) L(s)=((0.618−0.786i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(361s/2 ΓR(s+1) L(s)=((0.618−0.786i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 361361    =    19219^{2}
Sign: 0.618−0.786i0.618 - 0.786i
Analytic conductor: 38.794838.7948
Root analytic conductor: 38.794838.7948
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ361(72,⋅)\chi_{361} (72, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 361, (1: ), 0.618−0.786i)(1,\ 361,\ (1:\ ),\ 0.618 - 0.786i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.684881454−0.8186479329i1.684881454 - 0.8186479329i
L(12)L(\frac12) ≈\approx 1.684881454−0.8186479329i1.684881454 - 0.8186479329i
L(1)L(1) ≈\approx 1.126526105−0.3181898696i1.126526105 - 0.3181898696i
L(1)L(1) ≈\approx 1.126526105−0.3181898696i1.126526105 - 0.3181898696i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad19 1 1
good2 1+(0.435−0.900i)T 1 + (0.435 - 0.900i)T
3 1+(0.263+0.964i)T 1 + (0.263 + 0.964i)T
5 1+(−0.729−0.684i)T 1 + (-0.729 - 0.684i)T
7 1+(0.0275+0.999i)T 1 + (0.0275 + 0.999i)T
11 1+(0.350−0.936i)T 1 + (0.350 - 0.936i)T
13 1+(−0.418+0.908i)T 1 + (-0.418 + 0.908i)T
17 1+(−0.367−0.929i)T 1 + (-0.367 - 0.929i)T
23 1+(0.967−0.254i)T 1 + (0.967 - 0.254i)T
29 1+(0.979−0.200i)T 1 + (0.979 - 0.200i)T
31 1+(0.962−0.272i)T 1 + (0.962 - 0.272i)T
37 1+(0.986−0.164i)T 1 + (0.986 - 0.164i)T
41 1+(0.951+0.307i)T 1 + (0.951 + 0.307i)T
43 1+(−0.155−0.987i)T 1 + (-0.155 - 0.987i)T
47 1+(−0.649+0.760i)T 1 + (-0.649 + 0.760i)T
53 1+(0.333+0.942i)T 1 + (0.333 + 0.942i)T
59 1+(−0.209−0.977i)T 1 + (-0.209 - 0.977i)T
61 1+(−0.842−0.539i)T 1 + (-0.842 - 0.539i)T
67 1+(0.991+0.128i)T 1 + (0.991 + 0.128i)T
71 1+(0.842−0.539i)T 1 + (0.842 - 0.539i)T
73 1+(0.989+0.146i)T 1 + (0.989 + 0.146i)T
79 1+(0.777−0.628i)T 1 + (0.777 - 0.628i)T
83 1+(0.451−0.892i)T 1 + (0.451 - 0.892i)T
89 1+(−0.989+0.146i)T 1 + (-0.989 + 0.146i)T
97 1+(0.991−0.128i)T 1 + (0.991 - 0.128i)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

−24.56623701540944291132305834970, −23.70231662344816191060891819574, −22.9422305309409891572624125718, −22.68994828822150769968167207292, −21.2145015463555136401527900809, −19.88101404187648136863541708671, −19.51167443759334030366764777039, −18.10936097525073194450264866209, −17.5900006021725429103761339342, −16.745948051564338125267929590477, −15.32939345381061913669678946774, −14.81886745216537891317951162423, −13.97673875383192512679669464046, −12.97974393014678995188324143441, −12.33424276672206793232031915601, −11.19170928645802219235283432442, −9.902770680048915945129717195120, −8.39777934512047929366242132731, −7.70345994889200761595522523484, −6.97745152225105312912523981717, −6.34096892317540277347923729415, −4.76885292453383031327158512564, −3.71712221830111078047546419427, −2.70312539915284389326661872926, −0.80466701630539282122496744923, 0.65860567497577048519888264405, 2.41217669332751646146895267799, 3.30611472840788897295602740041, 4.48770203531716964973781174031, 4.99708375360308091720932219440, 6.217742774938936196467496498369, 8.215375453293684319540426653158, 9.07625903450656482199655008536, 9.49844588808037249982724256438, 11.03015145972155944982558507268, 11.57354477614724963400713417339, 12.354071520919667806177146889, 13.61518905624625853697844057016, 14.45844492612646149716670384684, 15.40867987074178674639026906163, 16.100054214493760652887375364462, 17.14298243157793258794285813184, 18.65543986945174404820544183264, 19.32938571112958815029034379019, 20.05294795015563047822222410364, 21.09831041910521253774736178817, 21.497226219996508373728617093504, 22.405687499623569018466560872827, 23.221331600489328474400437221195, 24.38911018403410250303215150175

Graph of the ZZ-function along the critical line