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Properties

Label 2-560-140.139-c3-0-0
Degree 22
Conductor 560560
Sign ii
Analytic cond. 33.041033.0410
Root an. cond. 5.748135.74813
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  + 5.29i·3-s − 11.1·5-s + 18.5i·7-s − 1.00·9-s + 11.8i·11-s − 40.2·13-s − 59.1i·15-s − 102.·17-s − 98.0·21-s + 125.·25-s + 137. i·27-s + 54·29-s − 62.6·33-s − 207. i·35-s − 212. i·39-s + ⋯
L(s)  = 1  + 1.01i·3-s − 0.999·5-s + 0.999i·7-s − 0.0370·9-s + 0.324i·11-s − 0.858·13-s − 1.01i·15-s − 1.46·17-s − 1.01·21-s + 1.00·25-s + 0.980i·27-s + 0.345·29-s − 0.330·33-s − 0.999i·35-s − 0.874i·39-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=(i Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+3/2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: ii
Analytic conductor: 33.041033.0410
Root analytic conductor: 5.748135.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ560(559,⋅)\chi_{560} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 560, ( :3/2), i)(2,\ 560,\ (\ :3/2),\ i)

Particular Values

L(2)L(2) ≈\approx 0.013816860270.01381686027
L(12)L(\frac12) ≈\approx 0.013816860270.01381686027
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
5 1+11.1T 1 + 11.1T
7 1−18.5iT 1 - 18.5iT
good3 1−5.29iT−27T2 1 - 5.29iT - 27T^{2}
11 1−11.8iT−1.33e3T2 1 - 11.8iT - 1.33e3T^{2}
13 1+40.2T+2.19e3T2 1 + 40.2T + 2.19e3T^{2}
17 1+102.T+4.91e3T2 1 + 102.T + 4.91e3T^{2}
19 1+6.85e3T2 1 + 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1−54T+2.43e4T2 1 - 54T + 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1−5.06e4T2 1 - 5.06e4T^{2}
41 1−6.89e4T2 1 - 6.89e4T^{2}
43 1+7.95e4T2 1 + 7.95e4T^{2}
47 1+619.iT−1.03e5T2 1 + 619. iT - 1.03e5T^{2}
53 1−1.48e5T2 1 - 1.48e5T^{2}
59 1+2.05e5T2 1 + 2.05e5T^{2}
61 1−2.26e5T2 1 - 2.26e5T^{2}
67 1+3.00e5T2 1 + 3.00e5T^{2}
71 1+863.iT−3.57e5T2 1 + 863. iT - 3.57e5T^{2}
73 1−523.T+3.89e5T2 1 - 523.T + 3.89e5T^{2}
79 1+1.38e3iT−4.93e5T2 1 + 1.38e3iT - 4.93e5T^{2}
83 1−47.6iT−5.71e5T2 1 - 47.6iT - 5.71e5T^{2}
89 1−7.04e5T2 1 - 7.04e5T^{2}
97 1−1.65e3T+9.12e5T2 1 - 1.65e3T + 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

−10.97446742269934774313160447588, −10.14191678023177708248960077131, −9.205631740162166647846577570091, −8.611266321821291651449529602339, −7.49535237374166221032348866420, −6.53952725778026332426293031947, −5.08620032099521976642001963032, −4.54004713186749754670027663461, −3.48260769112308358706784146607, −2.23358587901637427027205855716, 0.00466066659332570932309269716, 1.09192184039771230722238159706, 2.57192885542672481855826070835, 3.97615223445963869253852007286, 4.76311010377699498829224649719, 6.42663675288941001808518713762, 7.09184252326596840360865359561, 7.71408930089632715994406201013, 8.520840712004308781724209507971, 9.726757750335555876684513357505

Graph of the ZZ-function along the critical line