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Properties

Label 2-1152-72.43-c0-0-1
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(1087,⋅)\chi_{1152} (1087, \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 1.4302279661.430227966
L(12)L(\frac12) ≈\approx 1.4302279661.430227966
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

−9.838245322069321270051222710495, −9.245013408945647743742190859818, −8.351882239083700585824330099047, −7.959878493191411813236460292709, −6.66913407687611776171316649505, −5.86529393079910395565725270640, −4.71411806411841636565813913717, −3.71335588738781758439519472150, −3.06883693146696507906034106071, −1.63722745860050185923826561579, 1.60322794885275983020278715933, 2.49402314046716422586945968291, 3.84748085446877552486036163676, 4.47335688787094529705439088121, 5.94533847087419271686435575298, 6.82358281772922801825386637426, 7.44391732128144544139111138945, 8.354296547069605555751777138357, 9.050355540359003873534696919260, 9.878668284682313242443972409158

Graph of the ZZ-function along the critical line