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Properties

Label 2-60-20.7-c1-0-4
Degree 22
Conductor 6060
Sign 0.994−0.103i0.994 - 0.103i
Analytic cond. 0.4791020.479102
Root an. cond. 0.6921720.692172
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 0.394i)2-s + (−0.707 − 0.707i)3-s + (1.68 + 1.07i)4-s + (−1.75 − 1.38i)5-s + (−0.681 − 1.23i)6-s + (−2.47 + 2.47i)7-s + (1.87 + 2.11i)8-s + 1.00i·9-s + (−1.83 − 2.57i)10-s − 3.02i·11-s + (−0.437 − 1.95i)12-s + (0.363 − 0.363i)13-s + (−4.34 + 2.38i)14-s + (0.256 + 2.22i)15-s + (1.70 + 3.61i)16-s + (−2.36 − 2.36i)17-s + ⋯
L(s)  = 1  + (0.960 + 0.278i)2-s + (−0.408 − 0.408i)3-s + (0.844 + 0.535i)4-s + (−0.783 − 0.621i)5-s + (−0.278 − 0.505i)6-s + (−0.936 + 0.936i)7-s + (0.661 + 0.749i)8-s + 0.333i·9-s + (−0.579 − 0.814i)10-s − 0.913i·11-s + (−0.126 − 0.563i)12-s + (0.100 − 0.100i)13-s + (−1.16 + 0.638i)14-s + (0.0663 + 0.573i)15-s + (0.426 + 0.904i)16-s + (−0.573 − 0.573i)17-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.994−0.103i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1/2) L(s)=((0.994−0.103i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.994−0.103i0.994 - 0.103i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ60(7,⋅)\chi_{60} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 60, ( :1/2), 0.994−0.103i)(2,\ 60,\ (\ :1/2),\ 0.994 - 0.103i)

Particular Values

L(1)L(1) ≈\approx 1.13211+0.0588211i1.13211 + 0.0588211i
L(12)L(\frac12) ≈\approx 1.13211+0.0588211i1.13211 + 0.0588211i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−1.35−0.394i)T 1 + (-1.35 - 0.394i)T
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1+(1.75+1.38i)T 1 + (1.75 + 1.38i)T
good7 1+(2.47−2.47i)T−7iT2 1 + (2.47 - 2.47i)T - 7iT^{2}
11 1+3.02iT−11T2 1 + 3.02iT - 11T^{2}
13 1+(−0.363+0.363i)T−13iT2 1 + (-0.363 + 0.363i)T - 13iT^{2}
17 1+(2.36+2.36i)T+17iT2 1 + (2.36 + 2.36i)T + 17iT^{2}
19 1−4.95T+19T2 1 - 4.95T + 19T^{2}
23 1+(−0.900−0.900i)T+23iT2 1 + (-0.900 - 0.900i)T + 23iT^{2}
29 1−3.50iT−29T2 1 - 3.50iT - 29T^{2}
31 1+3.85iT−31T2 1 + 3.85iT - 31T^{2}
37 1+(0.363+0.363i)T+37iT2 1 + (0.363 + 0.363i)T + 37iT^{2}
41 1−2.72T+41T2 1 - 2.72T + 41T^{2}
43 1+(3.92+3.92i)T+43iT2 1 + (3.92 + 3.92i)T + 43iT^{2}
47 1+(5.85−5.85i)T−47iT2 1 + (5.85 - 5.85i)T - 47iT^{2}
53 1+(−3.14+3.14i)T−53iT2 1 + (-3.14 + 3.14i)T - 53iT^{2}
59 1+8.68T+59T2 1 + 8.68T + 59T^{2}
61 1+15.2T+61T2 1 + 15.2T + 61T^{2}
67 1+(−3.92+3.92i)T−67iT2 1 + (-3.92 + 3.92i)T - 67iT^{2}
71 1−4.25iT−71T2 1 - 4.25iT - 71T^{2}
73 1+(−9.28+9.28i)T−73iT2 1 + (-9.28 + 9.28i)T - 73iT^{2}
79 1+0.399T+79T2 1 + 0.399T + 79T^{2}
83 1+(−0.199−0.199i)T+83iT2 1 + (-0.199 - 0.199i)T + 83iT^{2}
89 1−4.28iT−89T2 1 - 4.28iT - 89T^{2}
97 1+(−6.73−6.73i)T+97iT2 1 + (-6.73 - 6.73i)T + 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−15.36790760819187953318912256821, −13.77488434072548104553198607366, −12.85022802355210355245258644241, −12.00324800912537837407477218774, −11.18726220073307894381233053579, −9.043323684358846287950424396288, −7.66632211036182986470108667259, −6.26972344492416972489918848919, −5.12045987168770241831574206311, −3.23476554326180866955618992189, 3.38477227989812797661300606691, 4.53011808042202940201052835960, 6.42737634161125575354669114117, 7.35642846402922270814853054122, 9.847185577527207281079643655937, 10.68976027580865864514972331879, 11.75249915797600334180292304126, 12.80726443676120092771426974029, 13.97020667312586228184528747230, 15.14842204528183520260688574443

Graph of the ZZ-function along the critical line