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Properties

Label 2-880-11.4-c1-0-5
Degree 22
Conductor 880880
Sign −0.278−0.960i-0.278 - 0.960i
Analytic cond. 7.026837.02683
Root an. cond. 2.650812.65081
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.584 + 1.79i)3-s + (0.809 − 0.587i)5-s + (0.846 + 2.60i)7-s + (−0.464 − 0.337i)9-s + (2.57 + 2.09i)11-s + (−0.159 − 0.115i)13-s + (0.584 + 1.79i)15-s + (1.18 − 0.862i)17-s + (0.828 − 2.55i)19-s − 5.18·21-s − 1.81·23-s + (0.309 − 0.951i)25-s + (−3.70 + 2.69i)27-s + (0.426 + 1.31i)29-s + (4.77 + 3.47i)31-s + ⋯
L(s)  = 1  + (−0.337 + 1.03i)3-s + (0.361 − 0.262i)5-s + (0.320 + 0.985i)7-s + (−0.154 − 0.112i)9-s + (0.775 + 0.631i)11-s + (−0.0441 − 0.0320i)13-s + (0.150 + 0.464i)15-s + (0.288 − 0.209i)17-s + (0.190 − 0.585i)19-s − 1.13·21-s − 0.378·23-s + (0.0618 − 0.190i)25-s + (−0.713 + 0.518i)27-s + (0.0791 + 0.243i)29-s + (0.857 + 0.623i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.278−0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((−0.278−0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.278−0.960i-0.278 - 0.960i
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ880(81,⋅)\chi_{880} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 880, ( :1/2), −0.278−0.960i)(2,\ 880,\ (\ :1/2),\ -0.278 - 0.960i)

Particular Values

L(1)L(1) ≈\approx 0.940617+1.25263i0.940617 + 1.25263i
L(12)L(\frac12) ≈\approx 0.940617+1.25263i0.940617 + 1.25263i
L(32)L(\frac{3}{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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
11 1+(−2.57−2.09i)T 1 + (-2.57 - 2.09i)T
good3 1+(0.584−1.79i)T+(−2.42−1.76i)T2 1 + (0.584 - 1.79i)T + (-2.42 - 1.76i)T^{2}
7 1+(−0.846−2.60i)T+(−5.66+4.11i)T2 1 + (-0.846 - 2.60i)T + (-5.66 + 4.11i)T^{2}
13 1+(0.159+0.115i)T+(4.01+12.3i)T2 1 + (0.159 + 0.115i)T + (4.01 + 12.3i)T^{2}
17 1+(−1.18+0.862i)T+(5.25−16.1i)T2 1 + (-1.18 + 0.862i)T + (5.25 - 16.1i)T^{2}
19 1+(−0.828+2.55i)T+(−15.3−11.1i)T2 1 + (-0.828 + 2.55i)T + (-15.3 - 11.1i)T^{2}
23 1+1.81T+23T2 1 + 1.81T + 23T^{2}
29 1+(−0.426−1.31i)T+(−23.4+17.0i)T2 1 + (-0.426 - 1.31i)T + (-23.4 + 17.0i)T^{2}
31 1+(−4.77−3.47i)T+(9.57+29.4i)T2 1 + (-4.77 - 3.47i)T + (9.57 + 29.4i)T^{2}
37 1+(−1.41−4.35i)T+(−29.9+21.7i)T2 1 + (-1.41 - 4.35i)T + (-29.9 + 21.7i)T^{2}
41 1+(−0.381+1.17i)T+(−33.1−24.0i)T2 1 + (-0.381 + 1.17i)T + (-33.1 - 24.0i)T^{2}
43 1+2.96T+43T2 1 + 2.96T + 43T^{2}
47 1+(2.87−8.84i)T+(−38.0−27.6i)T2 1 + (2.87 - 8.84i)T + (-38.0 - 27.6i)T^{2}
53 1+(2.17+1.58i)T+(16.3+50.4i)T2 1 + (2.17 + 1.58i)T + (16.3 + 50.4i)T^{2}
59 1+(4.39+13.5i)T+(−47.7+34.6i)T2 1 + (4.39 + 13.5i)T + (-47.7 + 34.6i)T^{2}
61 1+(6.10−4.43i)T+(18.8−58.0i)T2 1 + (6.10 - 4.43i)T + (18.8 - 58.0i)T^{2}
67 1+10.6T+67T2 1 + 10.6T + 67T^{2}
71 1+(5.56−4.04i)T+(21.9−67.5i)T2 1 + (5.56 - 4.04i)T + (21.9 - 67.5i)T^{2}
73 1+(−3.22−9.92i)T+(−59.0+42.9i)T2 1 + (-3.22 - 9.92i)T + (-59.0 + 42.9i)T^{2}
79 1+(−1.51−1.10i)T+(24.4+75.1i)T2 1 + (-1.51 - 1.10i)T + (24.4 + 75.1i)T^{2}
83 1+(4.68−3.40i)T+(25.6−78.9i)T2 1 + (4.68 - 3.40i)T + (25.6 - 78.9i)T^{2}
89 1−17.3T+89T2 1 - 17.3T + 89T^{2}
97 1+(−14.8−10.7i)T+(29.9+92.2i)T2 1 + (-14.8 - 10.7i)T + (29.9 + 92.2i)T^{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

−10.22242352725300243586082865221, −9.563682634942987861786994852993, −9.003404106710734047446493672222, −8.031266740289548539712979542768, −6.81970332910318741573927097501, −5.83649744510502977030674771442, −4.95888454824680121585557916007, −4.40440203865770197539505445159, −3.03984619614296848912605785134, −1.66486643679667480871358534222, 0.856620744019766690853499103171, 1.86512736383743533312782694323, 3.44832743202382635583048248939, 4.45562573045335775256237995505, 5.92175066278329085352374076180, 6.37459713062031873197095043381, 7.37096410842318092693508205289, 7.88192148154627029605891999973, 9.027039460008133525518226068041, 10.04531995094503149624469362810

Graph of the ZZ-function along the critical line