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Properties

Label 2-3960-1320.1139-c0-0-4
Degree 22
Conductor 39603960
Sign 0.422−0.906i0.422 - 0.906i
Analytic cond. 1.976291.97629
Root an. cond. 1.405801.40580
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.951 + 0.309i)5-s + (−1.16 − 1.59i)7-s + (−0.809 − 0.587i)8-s + 0.999i·10-s + (−0.156 + 0.987i)11-s + (0.863 − 0.280i)13-s + (1.16 − 1.59i)14-s + (0.309 − 0.951i)16-s + (0.363 − 0.5i)19-s + (−0.951 + 0.309i)20-s + (−0.987 + 0.156i)22-s + 1.61i·23-s + (0.809 + 0.587i)25-s + (0.533 + 0.734i)26-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.951 + 0.309i)5-s + (−1.16 − 1.59i)7-s + (−0.809 − 0.587i)8-s + 0.999i·10-s + (−0.156 + 0.987i)11-s + (0.863 − 0.280i)13-s + (1.16 − 1.59i)14-s + (0.309 − 0.951i)16-s + (0.363 − 0.5i)19-s + (−0.951 + 0.309i)20-s + (−0.987 + 0.156i)22-s + 1.61i·23-s + (0.809 + 0.587i)25-s + (0.533 + 0.734i)26-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.422−0.906i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((0.422−0.906i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.422−0.906i0.422 - 0.906i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(3779,⋅)\chi_{3960} (3779, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), 0.422−0.906i)(2,\ 3960,\ (\ :0),\ 0.422 - 0.906i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4817990631.481799063
L(12)L(\frac12) ≈\approx 1.4817990631.481799063
L(1)L(1) 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+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
3 1 1
5 1+(−0.951−0.309i)T 1 + (-0.951 - 0.309i)T
11 1+(0.156−0.987i)T 1 + (0.156 - 0.987i)T
good7 1+(1.16+1.59i)T+(−0.309+0.951i)T2 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2}
13 1+(−0.863+0.280i)T+(0.809−0.587i)T2 1 + (-0.863 + 0.280i)T + (0.809 - 0.587i)T^{2}
17 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
19 1+(−0.363+0.5i)T+(−0.309−0.951i)T2 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}
23 1−1.61iT−T2 1 - 1.61iT - T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(−1.44+1.04i)T+(0.309−0.951i)T2 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2}
41 1+(−0.253−0.183i)T+(0.309+0.951i)T2 1 + (-0.253 - 0.183i)T + (0.309 + 0.951i)T^{2}
43 1+T2 1 + T^{2}
47 1+(−1.11+1.53i)T+(−0.309−0.951i)T2 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2}
53 1+(−1.11+0.363i)T+(0.809−0.587i)T2 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2}
59 1+(−1.04−1.44i)T+(−0.309+0.951i)T2 1 + (-1.04 - 1.44i)T + (-0.309 + 0.951i)T^{2}
61 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
73 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
79 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
83 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
89 1+0.312iT−T2 1 + 0.312iT - T^{2}
97 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{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

−8.828339766243672067784300892253, −7.52973112452388742617903196878, −7.30333206183258453680634021742, −6.63148328503522691658231682721, −5.90917716695953143924542274681, −5.25623953817212222934631272855, −4.11705539995037436833043978138, −3.65365548105367691424483610897, −2.64522871970484653334397824677, −1.02397938545570086339597966092, 1.01855643797560794593288604239, 2.35030793761958260666208541378, 2.76661879373709806735809045412, 3.66057438486672440453985772199, 4.75169852463133925319726706989, 5.69802617454890655040955431756, 6.00916308767809429198804498962, 6.52688327775229899459714285694, 8.361599873605852928913109014502, 8.638992913954445298202071406176

Graph of the ZZ-function along the critical line