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Properties

Label 2-1152-24.5-c0-0-3
Degree $2$
Conductor $1152$
Sign $0.985 + 0.169i$
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  + 1.41·5-s − 2i·13-s + 1.41i·17-s + 1.00·25-s − 1.41·29-s + 1.41i·41-s − 49-s + 1.41·53-s − 2.82i·65-s + 2.00i·85-s − 1.41i·89-s − 1.41·101-s + 2i·109-s + 1.41i·113-s + ⋯
L(s)  = 1  + 1.41·5-s − 2i·13-s + 1.41i·17-s + 1.00·25-s − 1.41·29-s + 1.41i·41-s − 49-s + 1.41·53-s − 2.82i·65-s + 2.00i·85-s − 1.41i·89-s − 1.41·101-s + 2i·109-s + 1.41i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :0),\ 0.985 + 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.303016054\)
\(L(\frac12)\) \(\approx\) \(1.303016054\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.13181611503512820397641779331, −9.270507079497238495121685577367, −8.340749369997534379029070979007, −7.61844560403319871800897464167, −6.36037881166079769658999251801, −5.77914247163796537194159575314, −5.12333072742999378008992649051, −3.69224925260578778810051882166, −2.62381756048251923175897022981, −1.47995591810136245514367711367, 1.69588644847303352507954150731, 2.48919903996984099744771376927, 3.94250452564572894717490706497, 5.01147990142026700891968148701, 5.77975721144684164692655753849, 6.73283601041645292928138877343, 7.27303512395662761156427371385, 8.681260505457121391579601473073, 9.444885403718938626835519068909, 9.653715000013796772500004271944

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