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Properties

Label 2-3960-1320.1019-c0-0-0
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 0.20945883150.2094588315
L(12)L(\frac12) ≈\approx 0.20945883150.2094588315
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

−9.019136799713216576939688180772, −8.310407494116955437052255647923, −7.56307213494594231816729526532, −6.66848538917046507010642536221, −6.23074726420440841056421631294, −4.70860578515882443252111888416, −4.22537758967752172304825644814, −3.32180358494545687399796648446, −2.35127581059643719047896355209, −1.27931390574006041131103342001, 0.16056266344261937044968692791, 1.74067404911642514169791343972, 2.93970090292189434516186250297, 3.76636235021289157501804074846, 4.65659335861973577752992978082, 6.02472726072755092566824496461, 6.30884566524414452862082390681, 6.95945454846143795686522632534, 7.77675458890745773177105785751, 8.261600468495609636786059071521

Graph of the ZZ-function along the critical line