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Properties

Label 2-60-20.7-c1-0-2
Degree 22
Conductor 6060
Sign 0.742+0.669i0.742 + 0.669i
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  + (−1.41 + 0.0912i)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 + (−2.77 + 0.544i)8-s + 1.00i·9-s + (−1.69 + 2.66i)10-s + 0.728i·11-s + (−1.58 − 1.22i)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  + (−0.997 + 0.0645i)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 + (−0.981 + 0.192i)8-s + 0.333i·9-s + (−0.537 + 0.843i)10-s + 0.219i·11-s + (−0.457 − 0.352i)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)=((0.742+0.669i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1/2) L(s)=((0.742+0.669i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.742+0.669i0.742 + 0.669i
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), 0.742+0.669i)(2,\ 60,\ (\ :1/2),\ 0.742 + 0.669i)

Particular Values

L(1)L(1) ≈\approx 0.553334−0.212756i0.553334 - 0.212756i
L(12)L(\frac12) ≈\approx 0.553334−0.212756i0.553334 - 0.212756i
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+(1.41−0.0912i)T 1 + (1.41 - 0.0912i)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

−15.14139950590094773034654773369, −13.85607538287597096943185710776, −12.51889686939756867239089554308, −11.46157618837904673207840832693, −10.27664113391821455029778148176, −9.121596515668271958104885290191, −7.83933882858955501505071224003, −6.65568342474225312237821383992, −5.00239101024269964312071468942, −1.63912336428237542663196072666, 2.62726573415258421154532584130, 5.43689528568042674304442276403, 6.78861442324097595161681779551, 8.279486892703712789273985863842, 9.592237204499065363021444845433, 10.59496208992750199621730919392, 11.41324579748295637554493296223, 12.69957823024840685239536171254, 14.75710515582284237011383365522, 15.09425846566368733444020248830

Graph of the ZZ-function along the critical line