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Properties

Label 2-880-11.4-c1-0-3
Degree 22
Conductor 880880
Sign −0.202−0.979i-0.202 - 0.979i
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  + (−0.453 + 1.39i)3-s + (−0.809 + 0.587i)5-s + (−1.39 − 4.30i)7-s + (0.686 + 0.498i)9-s + (2.39 + 2.29i)11-s + (0.924 + 0.671i)13-s + (−0.453 − 1.39i)15-s + (−2.72 + 1.98i)17-s + (−1.88 + 5.78i)19-s + 6.63·21-s + 5.45·23-s + (0.309 − 0.951i)25-s + (−4.56 + 3.31i)27-s + (1.02 + 3.15i)29-s + (1.44 + 1.05i)31-s + ⋯
L(s)  = 1  + (−0.261 + 0.805i)3-s + (−0.361 + 0.262i)5-s + (−0.528 − 1.62i)7-s + (0.228 + 0.166i)9-s + (0.723 + 0.690i)11-s + (0.256 + 0.186i)13-s + (−0.117 − 0.360i)15-s + (−0.661 + 0.480i)17-s + (−0.431 + 1.32i)19-s + 1.44·21-s + 1.13·23-s + (0.0618 − 0.190i)25-s + (−0.878 + 0.638i)27-s + (0.190 + 0.586i)29-s + (0.260 + 0.189i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.202−0.979i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((−0.202−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.202−0.979i-0.202 - 0.979i
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), −0.202−0.979i)(2,\ 880,\ (\ :1/2),\ -0.202 - 0.979i)

Particular Values

L(1)L(1) ≈\approx 0.709280+0.870760i0.709280 + 0.870760i
L(12)L(\frac12) ≈\approx 0.709280+0.870760i0.709280 + 0.870760i
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+(−2.39−2.29i)T 1 + (-2.39 - 2.29i)T
good3 1+(0.453−1.39i)T+(−2.42−1.76i)T2 1 + (0.453 - 1.39i)T + (-2.42 - 1.76i)T^{2}
7 1+(1.39+4.30i)T+(−5.66+4.11i)T2 1 + (1.39 + 4.30i)T + (-5.66 + 4.11i)T^{2}
13 1+(−0.924−0.671i)T+(4.01+12.3i)T2 1 + (-0.924 - 0.671i)T + (4.01 + 12.3i)T^{2}
17 1+(2.72−1.98i)T+(5.25−16.1i)T2 1 + (2.72 - 1.98i)T + (5.25 - 16.1i)T^{2}
19 1+(1.88−5.78i)T+(−15.3−11.1i)T2 1 + (1.88 - 5.78i)T + (-15.3 - 11.1i)T^{2}
23 1−5.45T+23T2 1 - 5.45T + 23T^{2}
29 1+(−1.02−3.15i)T+(−23.4+17.0i)T2 1 + (-1.02 - 3.15i)T + (-23.4 + 17.0i)T^{2}
31 1+(−1.44−1.05i)T+(9.57+29.4i)T2 1 + (-1.44 - 1.05i)T + (9.57 + 29.4i)T^{2}
37 1+(−0.460−1.41i)T+(−29.9+21.7i)T2 1 + (-0.460 - 1.41i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.539−1.66i)T+(−33.1−24.0i)T2 1 + (0.539 - 1.66i)T + (-33.1 - 24.0i)T^{2}
43 1−0.263T+43T2 1 - 0.263T + 43T^{2}
47 1+(2.13−6.58i)T+(−38.0−27.6i)T2 1 + (2.13 - 6.58i)T + (-38.0 - 27.6i)T^{2}
53 1+(−1.16−0.846i)T+(16.3+50.4i)T2 1 + (-1.16 - 0.846i)T + (16.3 + 50.4i)T^{2}
59 1+(−2.18−6.72i)T+(−47.7+34.6i)T2 1 + (-2.18 - 6.72i)T + (-47.7 + 34.6i)T^{2}
61 1+(2.02−1.47i)T+(18.8−58.0i)T2 1 + (2.02 - 1.47i)T + (18.8 - 58.0i)T^{2}
67 1−0.516T+67T2 1 - 0.516T + 67T^{2}
71 1+(8.68−6.30i)T+(21.9−67.5i)T2 1 + (8.68 - 6.30i)T + (21.9 - 67.5i)T^{2}
73 1+(1.75+5.40i)T+(−59.0+42.9i)T2 1 + (1.75 + 5.40i)T + (-59.0 + 42.9i)T^{2}
79 1+(9.14+6.64i)T+(24.4+75.1i)T2 1 + (9.14 + 6.64i)T + (24.4 + 75.1i)T^{2}
83 1+(−3.62+2.63i)T+(25.6−78.9i)T2 1 + (-3.62 + 2.63i)T + (25.6 - 78.9i)T^{2}
89 1−13.2T+89T2 1 - 13.2T + 89T^{2}
97 1+(−2.71−1.97i)T+(29.9+92.2i)T2 1 + (-2.71 - 1.97i)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

−10.45870879694471657692181493507, −9.800268679659781827183791751134, −8.862183445993965381632815645531, −7.64398775975671412484791812426, −6.98883385897856957673560850466, −6.20281896338120886299136715956, −4.66028917387182569044913587951, −4.15735686105579514874714016108, −3.39251280480858933203336832357, −1.42695882419103868011706777773, 0.60360615455217618450959993120, 2.19486593281202721575727165502, 3.26849738626738037655165044410, 4.64160958235582135237800050585, 5.73222203072802227658650288230, 6.48626244131401933772536551886, 7.11639968555678693470783487906, 8.431999788801859243851558678151, 8.938461718918580435236757543045, 9.617327176926765097256217090683

Graph of the ZZ-function along the critical line