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Properties

Label 2-3960-1320.1019-c0-0-6
Degree 22
Conductor 39603960
Sign −0.100−0.994i-0.100 - 0.994i
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.863 − 0.280i)7-s + (−0.309 + 0.951i)8-s + i·10-s + (0.891 + 0.453i)11-s + (0.183 − 0.253i)13-s + (0.863 + 0.280i)14-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)19-s + (−0.587 + 0.809i)20-s + (0.453 + 0.891i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (0.297 − 0.0966i)26-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (0.863 − 0.280i)7-s + (−0.309 + 0.951i)8-s + i·10-s + (0.891 + 0.453i)11-s + (0.183 − 0.253i)13-s + (0.863 + 0.280i)14-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)19-s + (−0.587 + 0.809i)20-s + (0.453 + 0.891i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (0.297 − 0.0966i)26-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((−0.100−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((−0.100−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: −0.100−0.994i-0.100 - 0.994i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(2339,⋅)\chi_{3960} (2339, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), −0.100−0.994i)(2,\ 3960,\ (\ :0),\ -0.100 - 0.994i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5499471742.549947174
L(12)L(\frac12) ≈\approx 2.5499471742.549947174
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
3 1 1
5 1+(−0.587−0.809i)T 1 + (-0.587 - 0.809i)T
11 1+(−0.891−0.453i)T 1 + (-0.891 - 0.453i)T
good7 1+(−0.863+0.280i)T+(0.809−0.587i)T2 1 + (-0.863 + 0.280i)T + (0.809 - 0.587i)T^{2}
13 1+(−0.183+0.253i)T+(−0.309−0.951i)T2 1 + (-0.183 + 0.253i)T + (-0.309 - 0.951i)T^{2}
17 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
19 1+(1.53+0.5i)T+(0.809+0.587i)T2 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2}
23 1−0.618iT−T2 1 - 0.618iT - T^{2}
29 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(0.610+1.87i)T+(−0.809+0.587i)T2 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2}
41 1+(−0.550+1.69i)T+(−0.809−0.587i)T2 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2}
43 1+T2 1 + T^{2}
47 1+(−1.11−0.363i)T+(0.809+0.587i)T2 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2}
53 1+(−1.11+1.53i)T+(−0.309−0.951i)T2 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2}
59 1+(1.87−0.610i)T+(0.809−0.587i)T2 1 + (1.87 - 0.610i)T + (0.809 - 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1−T2 1 - T^{2}
71 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1−1.78iT−T2 1 - 1.78iT - T^{2}
97 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)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

−8.807491915289552295751793732338, −7.80423286775865219379145352274, −7.20234975922461442892803099946, −6.63827944296305866774618333215, −5.86833861869070588207112553280, −5.21486978240397544556660254663, −4.19599327187273566205726400798, −3.74038719501035370597211107948, −2.49039967657479490079090363752, −1.81178189391380518146111949468, 1.23222330980376701035337475041, 1.86257115170188356903929411334, 2.86937168367930113431151286147, 4.10856430617119427960983217967, 4.53631344044277202169194485122, 5.29776675753219828136171680578, 6.26481422649787733583005571691, 6.41570047444532325059317283578, 7.83514323841735608122372581779, 8.719694840354338502979960181854

Graph of the ZZ-function along the critical line