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Properties

Label 2-3072-96.53-c0-0-3
Degree 22
Conductor 30723072
Sign 0.195+0.980i0.195 + 0.980i
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
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  + (−0.923 + 0.382i)3-s + (0.541 + 0.541i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 1.70i)13-s + (−0.541 − 1.30i)19-s + (−0.707 − 0.292i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s − 1.84·31-s + (0.292 − 0.707i)37-s + (1.30 + 1.30i)39-s + (1.30 + 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (−1.70 + 0.707i)61-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.541 + 0.541i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 1.70i)13-s + (−0.541 − 1.30i)19-s + (−0.707 − 0.292i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s − 1.84·31-s + (0.292 − 0.707i)37-s + (1.30 + 1.30i)39-s + (1.30 + 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (−1.70 + 0.707i)61-s + ⋯

Functional equation

Λ(s)=(3072s/2 ΓC(s) L(s)=((0.195+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3072s/2 ΓC(s) L(s)=((0.195+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30723072    =    210⋅32^{10} \cdot 3
Sign: 0.195+0.980i0.195 + 0.980i
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3072(1409,⋅)\chi_{3072} (1409, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3072, ( :0), 0.195+0.980i)(2,\ 3072,\ (\ :0),\ 0.195 + 0.980i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66123791440.6612379144
L(12)L(\frac12) ≈\approx 0.66123791440.6612379144
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
3 1+(0.923−0.382i)T 1 + (0.923 - 0.382i)T
good5 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
7 1+(−0.541−0.541i)T+iT2 1 + (-0.541 - 0.541i)T + iT^{2}
11 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
13 1+(0.707+1.70i)T+(−0.707+0.707i)T2 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}
17 1+T2 1 + T^{2}
19 1+(0.541+1.30i)T+(−0.707+0.707i)T2 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2}
23 1+iT2 1 + iT^{2}
29 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
31 1+1.84T+T2 1 + 1.84T + T^{2}
37 1+(−0.292+0.707i)T+(−0.707−0.707i)T2 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(−1.30−0.541i)T+(0.707+0.707i)T2 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
59 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
61 1+(1.70−0.707i)T+(0.707−0.707i)T2 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2}
67 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
71 1−iT2 1 - iT^{2}
73 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
79 1+1.84iT−T2 1 + 1.84iT - T^{2}
83 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1−1.41T+T2 1 - 1.41T + 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

−8.895892851277243613738503742224, −7.81232223906855716815647088265, −7.32209781746216371024442291475, −6.22390029367628130793380713291, −5.60063527218153162192022841988, −4.99709570103875560136797441813, −4.24243479304154593103118792319, −3.10576577067648329037318044706, −2.06140395269720501797867106065, −0.45723127998124132989411076593, 1.47189524841912003188277277795, 2.11571277452877078173072909544, 3.84853687803841724122622008284, 4.39550369548282577741616021877, 5.28190116925457677210665824538, 6.03725789322019750956473829842, 6.85355828196574634868640017405, 7.44665924461593824074960436587, 8.048065270817418798924815667615, 9.197234810703116419437091083555

Graph of the ZZ-function along the critical line