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Properties

Label 2-1152-72.67-c0-0-0
Degree 22
Conductor 11521152
Sign 0.939+0.342i0.939 + 0.342i
Analytic cond. 0.5749220.574922
Root an. cond. 0.7582360.758236
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  + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)11-s + 17-s + 1.73·19-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (1.5 + 0.866i)33-s + (0.5 − 0.866i)41-s + (0.866 + 1.5i)43-s + (−0.5 + 0.866i)49-s + (−0.866 + 0.5i)51-s + (−1.49 + 0.866i)57-s + (0.866 − 1.5i)59-s + (0.866 − 1.5i)67-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)11-s + 17-s + 1.73·19-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (1.5 + 0.866i)33-s + (0.5 − 0.866i)41-s + (0.866 + 1.5i)43-s + (−0.5 + 0.866i)49-s + (−0.866 + 0.5i)51-s + (−1.49 + 0.866i)57-s + (0.866 − 1.5i)59-s + (0.866 − 1.5i)67-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s) L(s)=((0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 0.939+0.342i0.939 + 0.342i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1152(319,⋅)\chi_{1152} (319, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :0), 0.939+0.342i)(2,\ 1152,\ (\ :0),\ 0.939 + 0.342i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.77789840290.7778984029
L(12)L(\frac12) ≈\approx 0.77789840290.7778984029
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
good5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1−T+T2 1 - T + T^{2}
19 1−1.73T+T2 1 - 1.73T + T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T+T2 1 - T + T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
89 1+2T+T2 1 + 2T + T^{2}
97 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-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

−9.934693507884777883111692533549, −9.419622459537768662779858074983, −8.217630033926271213768049277123, −7.56441645998732547790818114098, −6.34009142564284027374456469336, −5.63194683064345890290452089239, −5.05263250295119888734388951383, −3.77225455404695305355969498953, −2.93578396386770892692027149065, −0.906721785050873677663317096060, 1.36436868098597528508970491646, 2.61970212251826839234668221787, 4.06298705114712841407454826840, 5.32646169398997055802109854777, 5.47712973199914126508290944881, 6.94447880615143813301849895722, 7.40545179070304723265444748795, 8.085807409644379365933909592063, 9.576779490568139547323868761367, 9.982934999379420077206598223379

Graph of the ZZ-function along the critical line