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Properties

Label 2-2268-252.83-c0-0-11
Degree 22
Conductor 22682268
Sign 0.996−0.0871i0.996 - 0.0871i
Analytic cond. 1.131871.13187
Root an. cond. 1.063891.06389
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.965 − 1.67i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (1.36 − 1.36i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)28-s + (1.22 + 0.707i)29-s + (0.258 + 0.965i)32-s − 1.73·37-s + (−0.866 − 0.5i)43-s + (1.67 − 0.965i)44-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.965 − 1.67i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (1.36 − 1.36i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)28-s + (1.22 + 0.707i)29-s + (0.258 + 0.965i)32-s − 1.73·37-s + (−0.866 − 0.5i)43-s + (1.67 − 0.965i)44-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.996−0.0871i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s) L(s)=((0.996−0.0871i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.996−0.0871i0.996 - 0.0871i
Analytic conductor: 1.131871.13187
Root analytic conductor: 1.063891.06389
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2268(755,⋅)\chi_{2268} (755, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :0), 0.996−0.0871i)(2,\ 2268,\ (\ :0),\ 0.996 - 0.0871i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1415917962.141591796
L(12)L(\frac12) ≈\approx 2.1415917962.141591796
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+(−0.965−0.258i)T 1 + (-0.965 - 0.258i)T
3 1 1
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good5 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
11 1+(−0.965+1.67i)T+(−0.5−0.866i)T2 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1+(−0.707−1.22i)T+(−0.5+0.866i)T2 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}
29 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+1.73T+T2 1 + 1.73T + T^{2}
41 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
43 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1−0.517iT−T2 1 - 0.517iT - T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
71 1+0.517T+T2 1 + 0.517T + T^{2}
73 1−T2 1 - T^{2}
79 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−8.941990447451365794792018377674, −8.531276389437033900506937774293, −7.35626057203883836739299039941, −6.72581306841658871676138480784, −6.12551543247300527703718737994, −5.35063748448071755361310606278, −4.30929973643031598907153571091, −3.38703637044892145761383098612, −3.03120966339614864431093644634, −1.30464371774011452369715669469, 1.55844927099671325418970651658, 2.58807203377252978256114020624, 3.44763913704678022493756135255, 4.45024606204535529815417843318, 5.02135961151817535200233376777, 6.11733956866806790981330344645, 6.80540282015095959401857991337, 7.17433497713015317158427177677, 8.540638266932095392567431516315, 9.378943711459767541135123132259

Graph of the ZZ-function along the critical line