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Properties

Label 2-1152-48.29-c2-0-6
Degree $2$
Conductor $1152$
Sign $0.435 - 0.900i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.84 − 1.84i)5-s + 0.226i·7-s + (11.5 + 11.5i)11-s + (8.11 + 8.11i)13-s − 6.20i·17-s + (−4.43 − 4.43i)19-s − 28.6·23-s − 18.1i·25-s + (−15.7 + 15.7i)29-s + 33.5·31-s + (0.418 − 0.418i)35-s + (−43.8 + 43.8i)37-s − 62.2·41-s + (18.3 − 18.3i)43-s + 13.8i·47-s + ⋯
L(s)  = 1  + (−0.369 − 0.369i)5-s + 0.0323i·7-s + (1.05 + 1.05i)11-s + (0.624 + 0.624i)13-s − 0.364i·17-s + (−0.233 − 0.233i)19-s − 1.24·23-s − 0.727i·25-s + (−0.543 + 0.543i)29-s + 1.08·31-s + (0.0119 − 0.0119i)35-s + (−1.18 + 1.18i)37-s − 1.51·41-s + (0.425 − 0.425i)43-s + 0.293i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.435 - 0.900i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.435 - 0.900i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.607119506\)
\(L(\frac12)\) \(\approx\) \(1.607119506\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (1.84 + 1.84i)T + 25iT^{2} \)
7 \( 1 - 0.226iT - 49T^{2} \)
11 \( 1 + (-11.5 - 11.5i)T + 121iT^{2} \)
13 \( 1 + (-8.11 - 8.11i)T + 169iT^{2} \)
17 \( 1 + 6.20iT - 289T^{2} \)
19 \( 1 + (4.43 + 4.43i)T + 361iT^{2} \)
23 \( 1 + 28.6T + 529T^{2} \)
29 \( 1 + (15.7 - 15.7i)T - 841iT^{2} \)
31 \( 1 - 33.5T + 961T^{2} \)
37 \( 1 + (43.8 - 43.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 62.2T + 1.68e3T^{2} \)
43 \( 1 + (-18.3 + 18.3i)T - 1.84e3iT^{2} \)
47 \( 1 - 13.8iT - 2.20e3T^{2} \)
53 \( 1 + (-37.9 - 37.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-70.4 - 70.4i)T + 3.48e3iT^{2} \)
61 \( 1 + (-60.3 - 60.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (-61.9 - 61.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 32.5T + 5.04e3T^{2} \)
73 \( 1 - 130. iT - 5.32e3T^{2} \)
79 \( 1 - 132.T + 6.24e3T^{2} \)
83 \( 1 + (19.2 - 19.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 91.2T + 7.92e3T^{2} \)
97 \( 1 + 20.0T + 9.40e3T^{2} \)
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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

−9.789330032031951867390966839847, −8.793066690854998038917546959271, −8.337673463696724595601225622955, −7.07995465611305788654349623422, −6.62575553550066433976465495662, −5.45726460750999122064224829641, −4.35300366708999818753897305359, −3.86376993451380155007383959182, −2.31079973511320415215185152117, −1.16530886231464704044862383339, 0.54637000571361546945931121764, 1.96182231111793233118477363418, 3.55463850691846583906879157717, 3.78527533012201939034027080540, 5.32399406698831724234158399180, 6.15872451857623394581497427814, 6.84941952050187128257689422905, 8.017519019844906220773326974245, 8.476109452847799912171319708237, 9.445195363553554726498781170050

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