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Properties

Label 2-560-140.139-c3-0-20
Degree 22
Conductor 560560
Sign −0.866−0.5i-0.866 - 0.5i
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  + 5.10i·3-s − 11.1·5-s + 18.5i·7-s + 0.987·9-s + 56.4i·11-s + 93.4·13-s − 57.0i·15-s + 133.·17-s − 94.4·21-s + 125.·25-s + 142. i·27-s − 293.·29-s − 287.·33-s − 207. i·35-s + 476. i·39-s + ⋯
L(s)  = 1  + 0.981i·3-s − 0.999·5-s + 0.999i·7-s + 0.0365·9-s + 1.54i·11-s + 1.99·13-s − 0.981i·15-s + 1.91·17-s − 0.981·21-s + 1.00·25-s + 1.01i·27-s − 1.87·29-s − 1.51·33-s − 0.999i·35-s + 1.95i·39-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=((−0.866−0.5i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+3/2) L(s)=((−0.866−0.5i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: −0.866−0.5i-0.866 - 0.5i
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), −0.866−0.5i)(2,\ 560,\ (\ :3/2),\ -0.866 - 0.5i)

Particular Values

L(2)L(2) ≈\approx 1.7731067121.773106712
L(12)L(\frac12) ≈\approx 1.7731067121.773106712
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 1−5.10iT−27T2 1 - 5.10iT - 27T^{2}
11 1−56.4iT−1.33e3T2 1 - 56.4iT - 1.33e3T^{2}
13 1−93.4T+2.19e3T2 1 - 93.4T + 2.19e3T^{2}
17 1−133.T+4.91e3T2 1 - 133.T + 4.91e3T^{2}
19 1+6.85e3T2 1 + 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1+293.T+2.43e4T2 1 + 293.T + 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 1−464.iT−1.03e5T2 1 - 464. 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 1−487.iT−4.93e5T2 1 - 487. iT - 4.93e5T^{2}
83 1−47.6iT−5.71e5T2 1 - 47.6iT - 5.71e5T^{2}
89 1−7.04e5T2 1 - 7.04e5T^{2}
97 1−8.91T+9.12e5T2 1 - 8.91T + 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

−10.75366304558648414557700993265, −9.754641005206173428733307805646, −9.127434462707278225831011542607, −8.113666150897531841887385373771, −7.31574256708600488001359512072, −5.96599745133750317687164246027, −5.01214383094717871361788175447, −3.98686749920909376786532335788, −3.30174515804012075183585366124, −1.50141352032545687751640243713, 0.64768688030830489541514491687, 1.28426465888549910823820143316, 3.48045641947450694057927250443, 3.78233804435459338053449594835, 5.57014327699056540694881045613, 6.46323392091214495684732360586, 7.46668484375992013001999538963, 8.007097840826949105235724318583, 8.756010834562195697058519201411, 10.21009306719603126530309114071

Graph of the ZZ-function along the critical line