Properties

Label 2-1152-24.5-c2-0-1
Degree $2$
Conductor $1152$
Sign $-0.816 - 0.577i$
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  − 5.83·5-s − 2.82·7-s + 16.4·11-s + 8.24i·13-s − 7.07i·17-s − 23.3i·19-s + 20i·23-s + 9·25-s + 29.1·29-s − 42.4·31-s + 16.4·35-s − 49.4i·37-s + 26.8i·41-s + 44i·47-s − 41·49-s + ⋯
L(s)  = 1  − 1.16·5-s − 0.404·7-s + 1.49·11-s + 0.634i·13-s − 0.415i·17-s − 1.22i·19-s + 0.869i·23-s + 0.359·25-s + 1.00·29-s − 1.36·31-s + 0.471·35-s − 1.33i·37-s + 0.655i·41-s + 0.936i·47-s − 0.836·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4281183330\)
\(L(\frac12)\) \(\approx\) \(0.4281183330\)
\(L(2)\) 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 \( 1 + 5.83T + 25T^{2} \)
7 \( 1 + 2.82T + 49T^{2} \)
11 \( 1 - 16.4T + 121T^{2} \)
13 \( 1 - 8.24iT - 169T^{2} \)
17 \( 1 + 7.07iT - 289T^{2} \)
19 \( 1 + 23.3iT - 361T^{2} \)
23 \( 1 - 20iT - 529T^{2} \)
29 \( 1 - 29.1T + 841T^{2} \)
31 \( 1 + 42.4T + 961T^{2} \)
37 \( 1 + 49.4iT - 1.36e3T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 44iT - 2.20e3T^{2} \)
53 \( 1 + 29.1T + 2.80e3T^{2} \)
59 \( 1 + 65.9T + 3.48e3T^{2} \)
61 \( 1 - 82.4iT - 3.72e3T^{2} \)
67 \( 1 - 116. iT - 4.48e3T^{2} \)
71 \( 1 - 100iT - 5.04e3T^{2} \)
73 \( 1 - 40T + 5.32e3T^{2} \)
79 \( 1 + 127.T + 6.24e3T^{2} \)
83 \( 1 + 82.4T + 6.88e3T^{2} \)
89 \( 1 - 114. iT - 7.92e3T^{2} \)
97 \( 1 + 40T + 9.40e3T^{2} \)
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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

−9.652601168274788452188659834102, −9.183993589310405047378928177709, −8.367706046993455774960941759792, −7.22436745420882560401070439960, −6.90325795711814257472123837115, −5.77937290448057761900684196649, −4.47318355644600606972721310072, −3.91624214104102468778459677922, −2.89617252801749625577047888618, −1.28450533022463371865202654589, 0.14159631471071321700838570546, 1.56790354967416291110838031234, 3.28691638143702816522427601846, 3.82507306757703508597721357712, 4.78470096863599948014877002952, 6.10175293087999770822942047413, 6.71548561198470993663807436580, 7.76188500382242069120518249136, 8.353602789521410727109781427557, 9.208329770558772901328377579432

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