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Properties

Label 2-880-11.3-c1-0-17
Degree 22
Conductor 880880
Sign 0.440+0.897i0.440 + 0.897i
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  + (−0.177 − 0.547i)3-s + (0.809 + 0.587i)5-s + (1.12 − 3.47i)7-s + (2.15 − 1.56i)9-s + (−0.490 + 3.28i)11-s + (2.29 − 1.66i)13-s + (0.177 − 0.547i)15-s + (−2.98 − 2.17i)17-s + (0.0293 + 0.0904i)19-s − 2.10·21-s − 1.16·23-s + (0.309 + 0.951i)25-s + (−2.63 − 1.91i)27-s + (−2.08 + 6.42i)29-s + (5.48 − 3.98i)31-s + ⋯
L(s)  = 1  + (−0.102 − 0.315i)3-s + (0.361 + 0.262i)5-s + (0.426 − 1.31i)7-s + (0.719 − 0.522i)9-s + (−0.147 + 0.989i)11-s + (0.635 − 0.461i)13-s + (0.0459 − 0.141i)15-s + (−0.724 − 0.526i)17-s + (0.00674 + 0.0207i)19-s − 0.458·21-s − 0.242·23-s + (0.0618 + 0.190i)25-s + (−0.507 − 0.369i)27-s + (−0.387 + 1.19i)29-s + (0.984 − 0.715i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((0.440+0.897i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((0.440+0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: 0.440+0.897i0.440 + 0.897i
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ880(641,⋅)\chi_{880} (641, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 880, ( :1/2), 0.440+0.897i)(2,\ 880,\ (\ :1/2),\ 0.440 + 0.897i)

Particular Values

L(1)L(1) ≈\approx 1.46676−0.914109i1.46676 - 0.914109i
L(12)L(\frac12) ≈\approx 1.46676−0.914109i1.46676 - 0.914109i
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+(0.490−3.28i)T 1 + (0.490 - 3.28i)T
good3 1+(0.177+0.547i)T+(−2.42+1.76i)T2 1 + (0.177 + 0.547i)T + (-2.42 + 1.76i)T^{2}
7 1+(−1.12+3.47i)T+(−5.66−4.11i)T2 1 + (-1.12 + 3.47i)T + (-5.66 - 4.11i)T^{2}
13 1+(−2.29+1.66i)T+(4.01−12.3i)T2 1 + (-2.29 + 1.66i)T + (4.01 - 12.3i)T^{2}
17 1+(2.98+2.17i)T+(5.25+16.1i)T2 1 + (2.98 + 2.17i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.0293−0.0904i)T+(−15.3+11.1i)T2 1 + (-0.0293 - 0.0904i)T + (-15.3 + 11.1i)T^{2}
23 1+1.16T+23T2 1 + 1.16T + 23T^{2}
29 1+(2.08−6.42i)T+(−23.4−17.0i)T2 1 + (2.08 - 6.42i)T + (-23.4 - 17.0i)T^{2}
31 1+(−5.48+3.98i)T+(9.57−29.4i)T2 1 + (-5.48 + 3.98i)T + (9.57 - 29.4i)T^{2}
37 1+(−3.04+9.35i)T+(−29.9−21.7i)T2 1 + (-3.04 + 9.35i)T + (-29.9 - 21.7i)T^{2}
41 1+(2.57+7.91i)T+(−33.1+24.0i)T2 1 + (2.57 + 7.91i)T + (-33.1 + 24.0i)T^{2}
43 1−2.96T+43T2 1 - 2.96T + 43T^{2}
47 1+(−0.687−2.11i)T+(−38.0+27.6i)T2 1 + (-0.687 - 2.11i)T + (-38.0 + 27.6i)T^{2}
53 1+(2.42−1.75i)T+(16.3−50.4i)T2 1 + (2.42 - 1.75i)T + (16.3 - 50.4i)T^{2}
59 1+(−2.62+8.09i)T+(−47.7−34.6i)T2 1 + (-2.62 + 8.09i)T + (-47.7 - 34.6i)T^{2}
61 1+(−6.86−4.98i)T+(18.8+58.0i)T2 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2}
67 1−13.4T+67T2 1 - 13.4T + 67T^{2}
71 1+(−6.71−4.88i)T+(21.9+67.5i)T2 1 + (-6.71 - 4.88i)T + (21.9 + 67.5i)T^{2}
73 1+(0.407−1.25i)T+(−59.0−42.9i)T2 1 + (0.407 - 1.25i)T + (-59.0 - 42.9i)T^{2}
79 1+(11.2−8.15i)T+(24.4−75.1i)T2 1 + (11.2 - 8.15i)T + (24.4 - 75.1i)T^{2}
83 1+(8.61+6.25i)T+(25.6+78.9i)T2 1 + (8.61 + 6.25i)T + (25.6 + 78.9i)T^{2}
89 1+12.1T+89T2 1 + 12.1T + 89T^{2}
97 1+(−3.50+2.54i)T+(29.9−92.2i)T2 1 + (-3.50 + 2.54i)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

−10.05199773497268518947514493141, −9.338167459705094174494984031632, −8.137513112604179252660591332017, −7.13316100776908285648171078709, −6.92960765574655250759325954404, −5.66875872823725525761419946800, −4.47782914483702712848969602528, −3.76598887600749613638144355664, −2.17257794492365013591415070593, −0.930607650645638655614440456585, 1.57615467347272262414299914567, 2.68833115532689616752372706878, 4.12604934046040278654595024825, 5.02915153711191487162313797489, 5.89255702661979190396490562115, 6.61297193975791982817119398896, 8.184762636900511430436484363782, 8.463988077818216626034623566206, 9.460514924610377323021031378390, 10.21385246144377640521972443392

Graph of the ZZ-function along the critical line