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Properties

Label 2-560-140.139-c3-0-13
Degree 22
Conductor 560560
Sign −0.812+0.583i-0.812 + 0.583i
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  + 7.05i·3-s + (9.69 + 5.57i)5-s + (−18.4 + 1.81i)7-s − 22.7·9-s + 22.9i·11-s + 35.9·13-s + (−39.3 + 68.3i)15-s − 80.0·17-s + 39.9·19-s + (−12.7 − 129. i)21-s − 183.·23-s + (62.8 + 108. i)25-s + 30.1i·27-s + 41.8·29-s − 223.·31-s + ⋯
L(s)  = 1  + 1.35i·3-s + (0.866 + 0.498i)5-s + (−0.995 + 0.0978i)7-s − 0.841·9-s + 0.629i·11-s + 0.766·13-s + (−0.676 + 1.17i)15-s − 1.14·17-s + 0.481·19-s + (−0.132 − 1.35i)21-s − 1.66·23-s + (0.502 + 0.864i)25-s + 0.214i·27-s + 0.268·29-s − 1.29·31-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=((−0.812+0.583i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+3/2) L(s)=((−0.812+0.583i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: −0.812+0.583i-0.812 + 0.583i
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), −0.812+0.583i)(2,\ 560,\ (\ :3/2),\ -0.812 + 0.583i)

Particular Values

L(2)L(2) ≈\approx 1.0390416751.039041675
L(12)L(\frac12) ≈\approx 1.0390416751.039041675
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+(−9.69−5.57i)T 1 + (-9.69 - 5.57i)T
7 1+(18.4−1.81i)T 1 + (18.4 - 1.81i)T
good3 1−7.05iT−27T2 1 - 7.05iT - 27T^{2}
11 1−22.9iT−1.33e3T2 1 - 22.9iT - 1.33e3T^{2}
13 1−35.9T+2.19e3T2 1 - 35.9T + 2.19e3T^{2}
17 1+80.0T+4.91e3T2 1 + 80.0T + 4.91e3T^{2}
19 1−39.9T+6.85e3T2 1 - 39.9T + 6.85e3T^{2}
23 1+183.T+1.21e4T2 1 + 183.T + 1.21e4T^{2}
29 1−41.8T+2.43e4T2 1 - 41.8T + 2.43e4T^{2}
31 1+223.T+2.97e4T2 1 + 223.T + 2.97e4T^{2}
37 1−91.3iT−5.06e4T2 1 - 91.3iT - 5.06e4T^{2}
41 1+138.iT−6.89e4T2 1 + 138. iT - 6.89e4T^{2}
43 1+169.T+7.95e4T2 1 + 169.T + 7.95e4T^{2}
47 1−92.1iT−1.03e5T2 1 - 92.1iT - 1.03e5T^{2}
53 1+258.iT−1.48e5T2 1 + 258. iT - 1.48e5T^{2}
59 1−250.T+2.05e5T2 1 - 250.T + 2.05e5T^{2}
61 1+65.9iT−2.26e5T2 1 + 65.9iT - 2.26e5T^{2}
67 1−658.T+3.00e5T2 1 - 658.T + 3.00e5T^{2}
71 1+370.iT−3.57e5T2 1 + 370. iT - 3.57e5T^{2}
73 1+511.T+3.89e5T2 1 + 511.T + 3.89e5T^{2}
79 1−26.3iT−4.93e5T2 1 - 26.3iT - 4.93e5T^{2}
83 1−1.12e3iT−5.71e5T2 1 - 1.12e3iT - 5.71e5T^{2}
89 1+1.59e3iT−7.04e5T2 1 + 1.59e3iT - 7.04e5T^{2}
97 1+927.T+9.12e5T2 1 + 927.T + 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

−10.59563910769091796477319892739, −9.928123177384822854310743429426, −9.466026925652555221092016784635, −8.616870321696890676355572072736, −7.07447824550760892053165695142, −6.21698498497414548104465184384, −5.34827945321897764483970442898, −4.15062026329563984041484443045, −3.32286625231057977898920329352, −2.04836286001859435659108897722, 0.29348300921991138364711009147, 1.48932457252497201326492026421, 2.52705876382220522634493251787, 3.93273027880068922353824425131, 5.59231856389937948043835319734, 6.26208272916152084687547642804, 6.89564883910688665950149276960, 8.063300362731704918605986537448, 8.865483191515453063513871129246, 9.716027704822262142706530562699

Graph of the ZZ-function along the critical line