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Properties

Label 2-3960-1320.899-c0-0-0
Degree 22
Conductor 39603960
Sign −0.971+0.237i-0.971 + 0.237i
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)=((−0.971+0.237i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((−0.971+0.237i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: −0.971+0.237i-0.971 + 0.237i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(899,⋅)\chi_{3960} (899, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), −0.971+0.237i)(2,\ 3960,\ (\ :0),\ -0.971 + 0.237i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.37129263970.3712926397
L(12)L(\frac12) ≈\approx 0.37129263970.3712926397
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

−8.650277075244991103433620553938, −8.263589857993992729027415109467, −7.72236390380761359128077773926, −6.88073483473396228661452871007, −6.40124269912689244484751120785, −5.35276453864687251337546242183, −4.74941231200308901410500295976, −3.67490702971968865524518876007, −2.42334180348627203894414741129, −1.71197993070622115989033890852, 0.26608817294915183836396236816, 1.49209345724541155130990750778, 2.47245181847088858780680448081, 3.52235878041276586429651351530, 4.40967626322896518115729220550, 4.95827560948643021991197545419, 6.10183693680003036126135862324, 7.10982091233073922205043550599, 7.85380603405360469009779961191, 8.372284000878403811603279691379

Graph of the ZZ-function along the critical line