Properties

Label 2-1152-8.5-c3-0-6
Degree $2$
Conductor $1152$
Sign $-i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3i·5-s − 14.6·7-s + 5.65i·11-s + 17·25-s − 218. i·29-s − 338.·31-s + 152. i·35-s − 127·49-s + 509. i·53-s + 58.7·55-s + 554. i·59-s + 322·73-s − 83.1i·77-s + 308.·79-s + 1.22e3i·83-s + ⋯
L(s)  = 1  − 0.929i·5-s − 0.793·7-s + 0.155i·11-s + 0.136·25-s − 1.39i·29-s − 1.95·31-s + 0.737i·35-s − 0.370·49-s + 1.31i·53-s + 0.144·55-s + 1.22i·59-s + 0.516·73-s − 0.123i·77-s + 0.439·79-s + 1.62i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6864387752\)
\(L(\frac12)\) \(\approx\) \(0.6864387752\)
\(L(\frac{5}{2})\) 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 \( 1 + 10.3iT - 125T^{2} \)
7 \( 1 + 14.6T + 343T^{2} \)
11 \( 1 - 5.65iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 218. iT - 2.43e4T^{2} \)
31 \( 1 + 338.T + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 509. iT - 1.48e5T^{2} \)
59 \( 1 - 554. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 322T + 3.89e5T^{2} \)
79 \( 1 - 308.T + 4.93e5T^{2} \)
83 \( 1 - 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 574T + 9.12e5T^{2} \)
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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

−9.441693097634920580878082873735, −8.994071311433229844906527274484, −8.024461881209783311379158359721, −7.18188474678392013337117004315, −6.19473252602108113973153109669, −5.39774657732659590211850382896, −4.43200511806724335986078499005, −3.52243760349827180664154857187, −2.28398610328640160690189559074, −0.982462127589477152619674389567, 0.18634918642795975289557131741, 1.84192360072070073976371195044, 3.08922685911511800726255743696, 3.61358475823917058532946860367, 4.99262636771099118916621861472, 5.96649615572181673186670565505, 6.81474034523309764541307555146, 7.31278632568474624046090893972, 8.458493780205957967859883984360, 9.304285163120311695607075791229

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