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Properties

Label 2-60-20.3-c1-0-2
Degree 22
Conductor 6060
Sign 0.546−0.837i0.546 - 0.837i
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  + (0.0912 + 1.41i)2-s + (0.707 − 0.707i)3-s + (−1.98 + 0.257i)4-s + (1.32 + 1.80i)5-s + (1.06 + 0.933i)6-s + (−1.86 − 1.86i)7-s + (−0.544 − 2.77i)8-s − 1.00i·9-s + (−2.42 + 2.02i)10-s + 0.728i·11-s + (−1.22 + 1.58i)12-s + (−3.12 − 3.12i)13-s + (2.46 − 2.80i)14-s + (2.20 + 0.342i)15-s + (3.86 − 1.02i)16-s + (1.12 − 1.12i)17-s + ⋯
L(s)  = 1  + (0.0645 + 0.997i)2-s + (0.408 − 0.408i)3-s + (−0.991 + 0.128i)4-s + (0.590 + 0.807i)5-s + (0.433 + 0.381i)6-s + (−0.705 − 0.705i)7-s + (−0.192 − 0.981i)8-s − 0.333i·9-s + (−0.767 + 0.641i)10-s + 0.219i·11-s + (−0.352 + 0.457i)12-s + (−0.866 − 0.866i)13-s + (0.658 − 0.749i)14-s + (0.570 + 0.0885i)15-s + (0.966 − 0.255i)16-s + (0.272 − 0.272i)17-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.546−0.837i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1/2) L(s)=((0.546−0.837i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.546−0.837i0.546 - 0.837i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ60(43,⋅)\chi_{60} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 60, ( :1/2), 0.546−0.837i)(2,\ 60,\ (\ :1/2),\ 0.546 - 0.837i)

Particular Values

L(1)L(1) ≈\approx 0.817228+0.442446i0.817228 + 0.442446i
L(12)L(\frac12) ≈\approx 0.817228+0.442446i0.817228 + 0.442446i
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+(−0.0912−1.41i)T 1 + (-0.0912 - 1.41i)T
3 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
5 1+(−1.32−1.80i)T 1 + (-1.32 - 1.80i)T
good7 1+(1.86+1.86i)T+7iT2 1 + (1.86 + 1.86i)T + 7iT^{2}
11 1−0.728iT−11T2 1 - 0.728iT - 11T^{2}
13 1+(3.12+3.12i)T+13iT2 1 + (3.12 + 3.12i)T + 13iT^{2}
17 1+(−1.12+1.12i)T−17iT2 1 + (-1.12 + 1.12i)T - 17iT^{2}
19 1−3.73T+19T2 1 - 3.73T + 19T^{2}
23 1+(5.83−5.83i)T−23iT2 1 + (5.83 - 5.83i)T - 23iT^{2}
29 1−2.64iT−29T2 1 - 2.64iT - 29T^{2}
31 1−6.01iT−31T2 1 - 6.01iT - 31T^{2}
37 1+(−3.12+3.12i)T−37iT2 1 + (-3.12 + 3.12i)T - 37iT^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 1+(−5.10+5.10i)T−43iT2 1 + (-5.10 + 5.10i)T - 43iT^{2}
47 1+(−2.09−2.09i)T+47iT2 1 + (-2.09 - 2.09i)T + 47iT^{2}
53 1+(−0.484−0.484i)T+53iT2 1 + (-0.484 - 0.484i)T + 53iT^{2}
59 1−4.92T+59T2 1 - 4.92T + 59T^{2}
61 1−2.31T+61T2 1 - 2.31T + 61T^{2}
67 1+(5.10+5.10i)T+67iT2 1 + (5.10 + 5.10i)T + 67iT^{2}
71 1+13.1iT−71T2 1 + 13.1iT - 71T^{2}
73 1+(−3.96−3.96i)T+73iT2 1 + (-3.96 - 3.96i)T + 73iT^{2}
79 1+7.11T+79T2 1 + 7.11T + 79T^{2}
83 1+(−3.55+3.55i)T−83iT2 1 + (-3.55 + 3.55i)T - 83iT^{2}
89 1−1.03iT−89T2 1 - 1.03iT - 89T^{2}
97 1+(12.5−12.5i)T−97iT2 1 + (12.5 - 12.5i)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

−15.16712187851486138980539456248, −14.12514395374100963679250615448, −13.52446230718183543050247813021, −12.34747804584616398179878238383, −10.23931636111382693776655762026, −9.474051717135374842054123744816, −7.67344560209680778661792403648, −6.96363737989232728965577444062, −5.57950822027377763077666186519, −3.37130257409424508583511936349, 2.45183576347053094532963864216, 4.35002774704167954760368500884, 5.80193236186153991877192141527, 8.347483012465051057578954155923, 9.456896137864312631782028417290, 9.978912696771910400214745553378, 11.74302201500702253236755527125, 12.60571447260429012613425372687, 13.66455272458973499926049386285, 14.61475605772709772827622503276

Graph of the ZZ-function along the critical line