Properties

Label 2-1152-72.67-c0-0-1
Degree $2$
Conductor $1152$
Sign $0.939 + 0.342i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.430227966\)
\(L(\frac12)\) \(\approx\) \(1.430227966\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line