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Properties

Label 2-3960-1320.299-c0-0-0
Degree 22
Conductor 39603960
Sign 0.906−0.422i0.906 - 0.422i
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)=((0.906−0.422i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((0.906−0.422i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.906−0.422i0.906 - 0.422i
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), 0.906−0.422i)(2,\ 3960,\ (\ :0),\ 0.906 - 0.422i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69125060360.6912506036
L(12)L(\frac12) ≈\approx 0.69125060360.6912506036
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

−8.888808749503127841960944785187, −7.974098848952819058176474405445, −7.31147094890603645384477614367, −6.43494707204278847466191842170, −5.51256337533275786362725426817, −4.68412916565345552088149929481, −4.04786849237894158086851065134, −3.12969268690269905203591080046, −2.56653201128035107726089009610, −1.19461142653265045354560645438, 0.38944782631777374307839999932, 2.26106392409611313642146388499, 3.54716606456850685437782663586, 4.18463209746546954438641750333, 4.73523336662415282796589926153, 5.56200474417411809437939569882, 6.77496352184355833491013535459, 6.92256955615487964255563942485, 7.70744705493643018121631340916, 8.555221430967833891226206552787

Graph of the ZZ-function along the critical line