↑

Properties

Label 2-60-20.3-c1-0-3
Degree 22
Conductor 6060
Sign 0.455+0.890i0.455 + 0.890i
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  + (−1.19 − 0.760i)2-s + (0.707 − 0.707i)3-s + (0.844 + 1.81i)4-s + (0.432 − 2.19i)5-s + (−1.38 + 0.305i)6-s + (−0.611 − 0.611i)7-s + (0.371 − 2.80i)8-s − 1.00i·9-s + (−2.18 + 2.28i)10-s + 5.12i·11-s + (1.87 + 0.685i)12-s + (1.76 + 1.76i)13-s + (0.264 + 1.19i)14-s + (−1.24 − 1.85i)15-s + (−2.57 + 3.06i)16-s + (−3.76 + 3.76i)17-s + ⋯
L(s)  = 1  + (−0.843 − 0.537i)2-s + (0.408 − 0.408i)3-s + (0.422 + 0.906i)4-s + (0.193 − 0.981i)5-s + (−0.563 + 0.124i)6-s + (−0.231 − 0.231i)7-s + (0.131 − 0.991i)8-s − 0.333i·9-s + (−0.690 + 0.723i)10-s + 1.54i·11-s + (0.542 + 0.197i)12-s + (0.488 + 0.488i)13-s + (0.0706 + 0.319i)14-s + (−0.321 − 0.479i)15-s + (−0.643 + 0.765i)16-s + (−0.912 + 0.912i)17-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.455+0.890i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1/2) L(s)=((0.455+0.890i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.455+0.890i0.455 + 0.890i
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), 0.455+0.890i)(2,\ 60,\ (\ :1/2),\ 0.455 + 0.890i)

Particular Values

L(1)L(1) ≈\approx 0.593269−0.362829i0.593269 - 0.362829i
L(12)L(\frac12) ≈\approx 0.593269−0.362829i0.593269 - 0.362829i
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+(1.19+0.760i)T 1 + (1.19 + 0.760i)T
3 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
5 1+(−0.432+2.19i)T 1 + (-0.432 + 2.19i)T
good7 1+(0.611+0.611i)T+7iT2 1 + (0.611 + 0.611i)T + 7iT^{2}
11 1−5.12iT−11T2 1 - 5.12iT - 11T^{2}
13 1+(−1.76−1.76i)T+13iT2 1 + (-1.76 - 1.76i)T + 13iT^{2}
17 1+(3.76−3.76i)T−17iT2 1 + (3.76 - 3.76i)T - 17iT^{2}
19 1−1.22T+19T2 1 - 1.22T + 19T^{2}
23 1+(−1.07+1.07i)T−23iT2 1 + (-1.07 + 1.07i)T - 23iT^{2}
29 1−0.864iT−29T2 1 - 0.864iT - 29T^{2}
31 1+7.81iT−31T2 1 + 7.81iT - 31T^{2}
37 1+(1.76−1.76i)T−37iT2 1 + (1.76 - 1.76i)T - 37iT^{2}
41 1−5.52T+41T2 1 - 5.52T + 41T^{2}
43 1+(6.20−6.20i)T−43iT2 1 + (6.20 - 6.20i)T - 43iT^{2}
47 1+(2.29+2.29i)T+47iT2 1 + (2.29 + 2.29i)T + 47iT^{2}
53 1+(2.62+2.62i)T+53iT2 1 + (2.62 + 2.62i)T + 53iT^{2}
59 1−0.528T+59T2 1 - 0.528T + 59T^{2}
61 1−4.98T+61T2 1 - 4.98T + 61T^{2}
67 1+(−6.20−6.20i)T+67iT2 1 + (-6.20 - 6.20i)T + 67iT^{2}
71 1+8.10iT−71T2 1 + 8.10iT - 71T^{2}
73 1+(2.25+2.25i)T+73iT2 1 + (2.25 + 2.25i)T + 73iT^{2}
79 1+15.9T+79T2 1 + 15.9T + 79T^{2}
83 1+(−7.95+7.95i)T−83iT2 1 + (-7.95 + 7.95i)T - 83iT^{2}
89 1−7.25iT−89T2 1 - 7.25iT - 89T^{2}
97 1+(−0.793+0.793i)T−97iT2 1 + (-0.793 + 0.793i)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

−15.07280915437925742761061930203, −13.28872046499776772258780600760, −12.72877450288727286117313601928, −11.58217129438887475327560269800, −10.01227946914180345037072447335, −9.110131533073862380999070467587, −8.026410503161687663804125371656, −6.69482606699020307559995109714, −4.22186681257837288851574594309, −1.87502238686416301347343536366, 2.99794960363558667981740078565, 5.64597234276793293522754012365, 6.89629736342839008472931529207, 8.336848530553082208476513259169, 9.352777320142061324897566286716, 10.61522855440015706435788908581, 11.32696421580510847118164097777, 13.61514662192035021469543101376, 14.33737008950654352740652010545, 15.60832160100677031397348988730

Graph of the ZZ-function along the critical line