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Properties

Label 2-1152-24.5-c4-0-47
Degree $2$
Conductor $1152$
Sign $0.985 + 0.169i$
Analytic cond. $119.082$
Root an. cond. $10.9124$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28.7·5-s + 80.4·7-s + 40.7·11-s + 36.7i·13-s − 108. i·17-s − 264. i·19-s + 406. i·23-s + 203.·25-s + 805.·29-s + 1.43e3·31-s + 2.31e3·35-s + 910. i·37-s − 2.44e3i·41-s − 3.54e3i·43-s + 1.74e3i·47-s + ⋯
L(s)  = 1  + 1.15·5-s + 1.64·7-s + 0.336·11-s + 0.217i·13-s − 0.374i·17-s − 0.732i·19-s + 0.768i·23-s + 0.326·25-s + 0.957·29-s + 1.49·31-s + 1.89·35-s + 0.665i·37-s − 1.45i·41-s − 1.91i·43-s + 0.787i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(119.082\)
Root analytic conductor: \(10.9124\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :2),\ 0.985 + 0.169i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.005354855\)
\(L(\frac12)\) \(\approx\) \(4.005354855\)
\(L(3)\) 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 \( 1 - 28.7T + 625T^{2} \)
7 \( 1 - 80.4T + 2.40e3T^{2} \)
11 \( 1 - 40.7T + 1.46e4T^{2} \)
13 \( 1 - 36.7iT - 2.85e4T^{2} \)
17 \( 1 + 108. iT - 8.35e4T^{2} \)
19 \( 1 + 264. iT - 1.30e5T^{2} \)
23 \( 1 - 406. iT - 2.79e5T^{2} \)
29 \( 1 - 805.T + 7.07e5T^{2} \)
31 \( 1 - 1.43e3T + 9.23e5T^{2} \)
37 \( 1 - 910. iT - 1.87e6T^{2} \)
41 \( 1 + 2.44e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.54e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.74e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.36e3T + 7.89e6T^{2} \)
59 \( 1 + 4.58e3T + 1.21e7T^{2} \)
61 \( 1 - 1.30e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.69e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.38e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.79e3T + 2.83e7T^{2} \)
79 \( 1 + 5.80e3T + 3.89e7T^{2} \)
83 \( 1 - 4.47e3T + 4.74e7T^{2} \)
89 \( 1 + 2.39e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.35e4T + 8.85e7T^{2} \)
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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

−9.113820240361189396772498141113, −8.555353870989179433511369132997, −7.57432445245267491857852792211, −6.72535655790667889057199715968, −5.70242633840107056236601791755, −5.02403741795611029624679938623, −4.20793685546380851447199926979, −2.67230864971398882596825994297, −1.81158723583958750445785750952, −0.956534788188922228540883995154, 1.07313283983394479120222574319, 1.78686101087809750235470494071, 2.77363503648930117996345010054, 4.29870784979323629801604894817, 4.96161971228539602401631034987, 5.92193820684761176241223171074, 6.56944206728235373294388860134, 7.918102697910434126570793396816, 8.289342760084690450231420146134, 9.295498283593430048305952042958

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