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Properties

Label 2-1152-8.5-c3-0-27
Degree 22
Conductor 11521152
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(4−s)\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}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=((0.707+0.707i) Λ‾(1−s)\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: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(577,⋅)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 0.707+0.707i)(2,\ 1152,\ (\ :3/2),\ 0.707 + 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.3528192521.352819252
L(12)L(\frac12) ≈\approx 1.3528192521.352819252
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
good5 1+12iT−125T2 1 + 12iT - 125T^{2}
7 1+32T+343T2 1 + 32T + 343T^{2}
11 1−8iT−1.33e3T2 1 - 8iT - 1.33e3T^{2}
13 1−20iT−2.19e3T2 1 - 20iT - 2.19e3T^{2}
17 1−98T+4.91e3T2 1 - 98T + 4.91e3T^{2}
19 1−88iT−6.85e3T2 1 - 88iT - 6.85e3T^{2}
23 1+32T+1.21e4T2 1 + 32T + 1.21e4T^{2}
29 1−172iT−2.43e4T2 1 - 172iT - 2.43e4T^{2}
31 1+256T+2.97e4T2 1 + 256T + 2.97e4T^{2}
37 1−92iT−5.06e4T2 1 - 92iT - 5.06e4T^{2}
41 1−102T+6.89e4T2 1 - 102T + 6.89e4T^{2}
43 1+296iT−7.95e4T2 1 + 296iT - 7.95e4T^{2}
47 1−320T+1.03e5T2 1 - 320T + 1.03e5T^{2}
53 1+76iT−1.48e5T2 1 + 76iT - 1.48e5T^{2}
59 1+408iT−2.05e5T2 1 + 408iT - 2.05e5T^{2}
61 1+636iT−2.26e5T2 1 + 636iT - 2.26e5T^{2}
67 1+552iT−3.00e5T2 1 + 552iT - 3.00e5T^{2}
71 1−416T+3.57e5T2 1 - 416T + 3.57e5T^{2}
73 1+138T+3.89e5T2 1 + 138T + 3.89e5T^{2}
79 1+64T+4.93e5T2 1 + 64T + 4.93e5T^{2}
83 1−392iT−5.71e5T2 1 - 392iT - 5.71e5T^{2}
89 1+582T+7.04e5T2 1 + 582T + 7.04e5T^{2}
97 1−238T+9.12e5T2 1 - 238T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line