↑

Properties

Label 2-60-20.3-c1-0-5
Degree 22
Conductor 6060
Sign 0.330+0.943i0.330 + 0.943i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.394 − 1.35i)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 + (−2.11 + 1.87i)8-s − 1.00i·9-s + (1.19 + 2.92i)10-s − 3.02i·11-s + (−1.95 + 0.437i)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  + (0.278 − 0.960i)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 + (−0.749 + 0.661i)8-s − 0.333i·9-s + (0.378 + 0.925i)10-s − 0.913i·11-s + (−0.563 + 0.126i)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)=((0.330+0.943i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1/2) L(s)=((0.330+0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.330+0.943i0.330 + 0.943i
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), 0.330+0.943i)(2,\ 60,\ (\ :1/2),\ 0.330 + 0.943i)

Particular Values

L(1)L(1) ≈\approx 0.792001−0.561615i0.792001 - 0.561615i
L(12)L(\frac12) ≈\approx 0.792001−0.561615i0.792001 - 0.561615i
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+(−0.394+1.35i)T 1 + (-0.394 + 1.35i)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}
show more
show less
   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

−14.74341861826813309516692583027, −13.74840395062219573683379365608, −12.46900599999743916059175353573, −11.50594775644716363232487736596, −10.74649585344510716254481436925, −8.897687794369621347888782138137, −8.073896344998307621880360215356, −6.01914623310028412066079288851, −4.10845597245463350886600356084, −2.42227100475349896467561560502, 4.10820850581918588614915726140, 4.86935030706652868948885848305, 7.10037216449274047057385336481, 8.059719745717866554026791003454, 9.084434006859695675261731118522, 10.70336046015381301690921460705, 12.23285714176562024623902312620, 13.37490385790743472992871812497, 14.49378797098679069970408126560, 15.25500950041580789057019742191

Graph of the ZZ-function along the critical line