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Properties

Label 2-1152-8.5-c3-0-21
Degree $2$
Conductor $1152$
Sign $0.707 - 0.707i$
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  + 12i·5-s − 32·7-s − 8i·11-s − 20i·13-s + 98·17-s − 88i·19-s − 32·23-s − 19·25-s − 172i·29-s − 256·31-s − 384i·35-s − 92i·37-s + 102·41-s + 296i·43-s + 320·47-s + ⋯
L(s)  = 1  + 1.07i·5-s − 1.72·7-s − 0.219i·11-s − 0.426i·13-s + 1.39·17-s − 1.06i·19-s − 0.290·23-s − 0.151·25-s − 1.10i·29-s − 1.48·31-s − 1.85i·35-s − 0.408i·37-s + 0.388·41-s + 1.04i·43-s + 0.993·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1152,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.352819252\)
\(L(\frac12)\) \(\approx\) \(1.352819252\)
\(L(\frac{5}{2})\) 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 \( 1 - 12iT - 125T^{2} \)
7 \( 1 + 32T + 343T^{2} \)
11 \( 1 + 8iT - 1.33e3T^{2} \)
13 \( 1 + 20iT - 2.19e3T^{2} \)
17 \( 1 - 98T + 4.91e3T^{2} \)
19 \( 1 + 88iT - 6.85e3T^{2} \)
23 \( 1 + 32T + 1.21e4T^{2} \)
29 \( 1 + 172iT - 2.43e4T^{2} \)
31 \( 1 + 256T + 2.97e4T^{2} \)
37 \( 1 + 92iT - 5.06e4T^{2} \)
41 \( 1 - 102T + 6.89e4T^{2} \)
43 \( 1 - 296iT - 7.95e4T^{2} \)
47 \( 1 - 320T + 1.03e5T^{2} \)
53 \( 1 - 76iT - 1.48e5T^{2} \)
59 \( 1 - 408iT - 2.05e5T^{2} \)
61 \( 1 - 636iT - 2.26e5T^{2} \)
67 \( 1 - 552iT - 3.00e5T^{2} \)
71 \( 1 - 416T + 3.57e5T^{2} \)
73 \( 1 + 138T + 3.89e5T^{2} \)
79 \( 1 + 64T + 4.93e5T^{2} \)
83 \( 1 + 392iT - 5.71e5T^{2} \)
89 \( 1 + 582T + 7.04e5T^{2} \)
97 \( 1 - 238T + 9.12e5T^{2} \)
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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

−9.666664658809018361517647869113, −8.890809329735441799240492357002, −7.58914452445000718275572054090, −7.07104639519245952422796475666, −6.15714614460348207807664741314, −5.60699025300210885792316331158, −4.00312819470748420122643950859, −3.15367373787743586084644408835, −2.59077756564567735159746202525, −0.67341887266093627643859601061, 0.52020198516688606631277704698, 1.74096922812398386298285286014, 3.25915613963670719844321856028, 3.88892057072972161905692027248, 5.17957456048342191021504473275, 5.85967078359538650728688935205, 6.79746673901580801844728352654, 7.66418723613323904447223764772, 8.674862319785661612836621078538, 9.392656189466090410047124048107

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