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Properties

Label 2-60-60.59-c1-0-5
Degree 22
Conductor 6060
Sign 0.968−0.250i0.968 - 0.250i
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  + (1.11 + 0.866i)2-s − 1.73i·3-s + (0.500 + 1.93i)4-s − 2.23·5-s + (1.49 − 1.93i)6-s + (−1.11 + 2.59i)8-s − 2.99·9-s + (−2.50 − 1.93i)10-s + (3.35 − 0.866i)12-s + 3.87i·15-s + (−3.5 + 1.93i)16-s + 4.47·17-s + (−3.35 − 2.59i)18-s − 7.74i·19-s + (−1.11 − 4.33i)20-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)2-s − 0.999i·3-s + (0.250 + 0.968i)4-s − 0.999·5-s + (0.612 − 0.790i)6-s + (−0.395 + 0.918i)8-s − 0.999·9-s + (−0.790 − 0.612i)10-s + (0.968 − 0.250i)12-s + 1.00i·15-s + (−0.875 + 0.484i)16-s + 1.08·17-s + (−0.790 − 0.612i)18-s − 1.77i·19-s + (−0.250 − 0.968i)20-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.968−0.250i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1/2) L(s)=((0.968−0.250i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.968−0.250i0.968 - 0.250i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ60(59,⋅)\chi_{60} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 60, ( :1/2), 0.968−0.250i)(2,\ 60,\ (\ :1/2),\ 0.968 - 0.250i)

Particular Values

L(1)L(1) ≈\approx 1.10189+0.139958i1.10189 + 0.139958i
L(12)L(\frac12) ≈\approx 1.10189+0.139958i1.10189 + 0.139958i
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+(−1.11−0.866i)T 1 + (-1.11 - 0.866i)T
3 1+1.73iT 1 + 1.73iT
5 1+2.23T 1 + 2.23T
good7 1+7T2 1 + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1−4.47T+17T2 1 - 4.47T + 17T^{2}
19 1+7.74iT−19T2 1 + 7.74iT - 19T^{2}
23 1−3.46iT−23T2 1 - 3.46iT - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1−7.74iT−31T2 1 - 7.74iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+10.3iT−47T2 1 + 10.3iT - 47T^{2}
53 1+4.47T+53T2 1 + 4.47T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−7.74iT−79T2 1 - 7.74iT - 79T^{2}
83 1+3.46iT−83T2 1 + 3.46iT - 83T^{2}
89 1−89T2 1 - 89T^{2}
97 1−97T2 1 - 97T^{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

−15.05036557585281203836587496349, −14.01906401124226464397554511700, −12.96969218899520038325655249781, −12.05063592837146378367776714785, −11.21786645245511680209390523208, −8.710604288240186882615011007122, −7.62130075998172126003502019318, −6.75590916740167205077218524161, −5.11991821723020958559461200269, −3.21612165045091494677779403163, 3.34562197376954970251902214197, 4.44720620051296434479123125516, 5.91230804043502411888542382903, 7.997428208638764822405330422004, 9.652140086505478420732511583620, 10.66466370514893032831262775356, 11.69263790237918546626704841057, 12.57248072451275670816841051627, 14.27361460814636433910119904715, 14.85454371831900041373055145068

Graph of the ZZ-function along the critical line