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Properties

Label 2-3872-88.3-c0-0-1
Degree 22
Conductor 38723872
Sign 0.970+0.242i0.970 + 0.242i
Analytic cond. 1.932371.93237
Root an. cond. 1.390101.39010
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  + (−0.190 − 0.587i)3-s + (0.5 − 0.363i)9-s + (1.30 + 0.951i)17-s + (0.5 + 1.53i)19-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.190 + 0.587i)41-s − 0.618·43-s + (−0.809 − 0.587i)49-s + (0.309 − 0.951i)51-s + (0.809 − 0.587i)57-s + (0.5 − 1.53i)59-s + 1.61·67-s + (0.190 − 0.587i)73-s + (0.5 − 0.363i)75-s + ⋯
L(s)  = 1  + (−0.190 − 0.587i)3-s + (0.5 − 0.363i)9-s + (1.30 + 0.951i)17-s + (0.5 + 1.53i)19-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.190 + 0.587i)41-s − 0.618·43-s + (−0.809 − 0.587i)49-s + (0.309 − 0.951i)51-s + (0.809 − 0.587i)57-s + (0.5 − 1.53i)59-s + 1.61·67-s + (0.190 − 0.587i)73-s + (0.5 − 0.363i)75-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((0.970+0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s) L(s)=((0.970+0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 0.970+0.242i0.970 + 0.242i
Analytic conductor: 1.932371.93237
Root analytic conductor: 1.390101.39010
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3872(1455,⋅)\chi_{3872} (1455, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :0), 0.970+0.242i)(2,\ 3872,\ (\ :0),\ 0.970 + 0.242i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3410641161.341064116
L(12)L(\frac12) ≈\approx 1.3410641161.341064116
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 1
11 1 1
good3 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}
5 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
7 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
13 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
17 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
19 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
23 1−T2 1 - 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+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
41 1+(−0.190−0.587i)T+(−0.809+0.587i)T2 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}
43 1+0.618T+T2 1 + 0.618T + T^{2}
47 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
53 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
59 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
61 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
67 1−1.61T+T2 1 - 1.61T + T^{2}
71 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
73 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
79 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
83 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
89 1−0.618T+T2 1 - 0.618T + T^{2}
97 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (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

−8.326050721334343529810883974680, −7.935421209579347632186535467520, −7.17524784941123476321920731668, −6.41975433485721405781453964527, −5.76508775386321506666672070296, −5.02472956278491721554359952515, −3.78971209761155360699784115861, −3.37274262229369935377825936310, −1.87176521693342426096436681134, −1.17838969455477199886323253389, 0.981682183584508824880832420051, 2.39669458012458752597583341944, 3.24805215846863542836296870444, 4.22374558718100441281771083578, 4.98562081994747938582186822852, 5.43470860799048398654561244468, 6.55322448122963253208779282278, 7.26198679935756144741850182064, 7.86118758419897716921171784890, 8.809581134972902504610123131467

Graph of the ZZ-function along the critical line