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Properties

Label 1-185-185.174-r0-0-0
Degree 11
Conductor 185185
Sign 0.989−0.146i0.989 - 0.146i
Analytic cond. 0.8591360.859136
Root an. cond. 0.8591360.859136
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + 11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + 11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

Λ(s)=(185s/2 ΓR(s) L(s)=((0.989−0.146i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(185s/2 ΓR(s) L(s)=((0.989−0.146i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 185185    =    5⋅375 \cdot 37
Sign: 0.989−0.146i0.989 - 0.146i
Analytic conductor: 0.8591360.859136
Root analytic conductor: 0.8591360.859136
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ185(174,⋅)\chi_{185} (174, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 185, (0: ), 0.989−0.146i)(1,\ 185,\ (0:\ ),\ 0.989 - 0.146i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.702033464−0.1254126183i1.702033464 - 0.1254126183i
L(12)L(\frac12) ≈\approx 1.702033464−0.1254126183i1.702033464 - 0.1254126183i
L(1)L(1) ≈\approx 1.507527745−0.1821017226i1.507527745 - 0.1821017226i
L(1)L(1) ≈\approx 1.507527745−0.1821017226i1.507527745 - 0.1821017226i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
37 1 1
good2 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
11 1+T 1 + T
13 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
17 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
23 1−T 1 - T
29 1+T 1 + T
31 1+T 1 + T
41 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
43 1−T 1 - T
47 1−T 1 - T
53 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
59 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
61 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
67 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
71 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
73 1−T 1 - T
79 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
83 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
89 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
97 1−T 1 - 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

−26.95081321895478660784268020368, −26.000546179149819179692946957331, −25.14470735763668068859913360207, −24.48877325757474730769236405047, −23.42766609899855429515570166456, −22.98649778663084207034998058089, −21.55882349769026327196977884732, −20.50761991150139907060316595300, −19.57594728176509500370819087728, −18.26978439621610670775176737910, −17.43553986341678199874609899235, −16.66611151341352274235132605161, −15.18339231403320462183155591109, −14.35219811523082985326069712073, −13.6825737069693807628285928410, −12.65215627408378675734905074695, −11.73852630007923337147584208714, −10.05176443214779254377092545588, −8.37137974366604994558023956682, −8.017140058566335578019757346853, −6.72471948224281402971426144084, −5.948069925076772901157372874477, −4.2504514158167553772184910392, −3.29302316214225791655031664153, −1.37727858296318616570241455100, 1.81223489548653750515351876897, 2.97164868000308392516364351015, 4.19532900827566200343217661069, 5.037073156462614083561782061873, 6.36434392654950458235586062216, 8.48796833718866774248952604171, 9.18337656873534024999112632682, 10.16304783268726934643677184533, 11.467373928444306553375914838273, 11.94612721090948557760751803714, 13.62527632701931374458932585165, 14.30467547735306003203453175297, 15.18799872183317969590322671115, 16.18794557570385903717365843196, 17.66789873692879982659758722052, 18.86397853758001236930707694762, 19.66093414696688923397476292280, 20.664348454008886533701196414011, 21.486848247244692962189185947567, 22.013228581721399475144718327305, 23.068289868910321049896634294068, 24.315167222352474304242662947657, 25.24671685478114539420224227461, 26.44308857699347374942006622692, 27.60388540250650980742254007732

Graph of the ZZ-function along the critical line