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Properties

Label 2-3960-1320.299-c0-0-4
Degree 22
Conductor 39603960
Sign 0.700+0.713i0.700 + 0.713i
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 + (−0.183 + 0.253i)7-s + (0.809 − 0.587i)8-s + 0.999i·10-s + (−0.987 + 0.156i)11-s + (−1.69 − 0.550i)13-s + (−0.183 − 0.253i)14-s + (0.309 + 0.951i)16-s + (−0.363 − 0.5i)19-s + (−0.951 − 0.309i)20-s + (0.156 − 0.987i)22-s − 1.61i·23-s + (0.809 − 0.587i)25-s + (1.04 − 1.44i)26-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (−0.183 + 0.253i)7-s + (0.809 − 0.587i)8-s + 0.999i·10-s + (−0.987 + 0.156i)11-s + (−1.69 − 0.550i)13-s + (−0.183 − 0.253i)14-s + (0.309 + 0.951i)16-s + (−0.363 − 0.5i)19-s + (−0.951 − 0.309i)20-s + (0.156 − 0.987i)22-s − 1.61i·23-s + (0.809 − 0.587i)25-s + (1.04 − 1.44i)26-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.700+0.713i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((0.700+0.713i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.700+0.713i0.700 + 0.713i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(1619,⋅)\chi_{3960} (1619, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), 0.700+0.713i)(2,\ 3960,\ (\ :0),\ 0.700 + 0.713i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.72468866840.7246886684
L(12)L(\frac12) ≈\approx 0.72468866840.7246886684
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+(0.987−0.156i)T 1 + (0.987 - 0.156i)T
good7 1+(0.183−0.253i)T+(−0.309−0.951i)T2 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2}
13 1+(1.69+0.550i)T+(0.809+0.587i)T2 1 + (1.69 + 0.550i)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+(−0.734−0.533i)T+(0.309+0.951i)T2 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2}
41 1+(−1.59+1.16i)T+(0.309−0.951i)T2 1 + (-1.59 + 1.16i)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+(−0.533+0.734i)T+(−0.309−0.951i)T2 1 + (-0.533 + 0.734i)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+1.97iT−T2 1 + 1.97iT - 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

−8.482574747722734675760319290621, −7.82117781790747488862686464160, −7.05719413122585773083628233118, −6.36662764978008248958202451359, −5.58948555045842812317350395169, −4.96964983808520353156904825269, −4.46412963115344089580883952733, −2.80091412680833316876344820724, −2.12352777116065178785801289378, −0.43695981798257153839927964775, 1.43214921403448145097296640256, 2.42499591466296938701961123239, 2.90754120762772124436827162102, 4.04989958509836348092191642003, 4.94217013518638586312485494492, 5.56082314087060207625491007027, 6.56182604501615659225934368215, 7.62069204225928416710377310953, 7.83356033690376033295804298731, 9.128175038588413475063138372960

Graph of the ZZ-function along the critical line