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Properties

Label 2-880-11.4-c1-0-23
Degree 22
Conductor 880880
Sign −0.939+0.343i-0.939 + 0.343i
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  + (1.01 − 3.13i)3-s + (0.809 − 0.587i)5-s + (−1.08 − 3.34i)7-s + (−6.34 − 4.61i)9-s + (1.91 − 2.70i)11-s + (3.45 + 2.51i)13-s + (−1.01 − 3.13i)15-s + (−1.28 + 0.934i)17-s + (−1.71 + 5.27i)19-s − 11.5·21-s + 6.39·23-s + (0.309 − 0.951i)25-s + (−12.9 + 9.37i)27-s + (−0.117 − 0.362i)29-s + (−0.615 − 0.446i)31-s + ⋯
L(s)  = 1  + (0.587 − 1.80i)3-s + (0.361 − 0.262i)5-s + (−0.410 − 1.26i)7-s + (−2.11 − 1.53i)9-s + (0.577 − 0.816i)11-s + (0.958 + 0.696i)13-s + (−0.262 − 0.808i)15-s + (−0.311 + 0.226i)17-s + (−0.393 + 1.21i)19-s − 2.52·21-s + 1.33·23-s + (0.0618 − 0.190i)25-s + (−2.48 + 1.80i)27-s + (−0.0218 − 0.0673i)29-s + (−0.110 − 0.0802i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.939+0.343i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((−0.939+0.343i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.939+0.343i-0.939 + 0.343i
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), −0.939+0.343i)(2,\ 880,\ (\ :1/2),\ -0.939 + 0.343i)

Particular Values

L(1)L(1) ≈\approx 0.328367−1.85325i0.328367 - 1.85325i
L(12)L(\frac12) ≈\approx 0.328367−1.85325i0.328367 - 1.85325i
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+(−1.91+2.70i)T 1 + (-1.91 + 2.70i)T
good3 1+(−1.01+3.13i)T+(−2.42−1.76i)T2 1 + (-1.01 + 3.13i)T + (-2.42 - 1.76i)T^{2}
7 1+(1.08+3.34i)T+(−5.66+4.11i)T2 1 + (1.08 + 3.34i)T + (-5.66 + 4.11i)T^{2}
13 1+(−3.45−2.51i)T+(4.01+12.3i)T2 1 + (-3.45 - 2.51i)T + (4.01 + 12.3i)T^{2}
17 1+(1.28−0.934i)T+(5.25−16.1i)T2 1 + (1.28 - 0.934i)T + (5.25 - 16.1i)T^{2}
19 1+(1.71−5.27i)T+(−15.3−11.1i)T2 1 + (1.71 - 5.27i)T + (-15.3 - 11.1i)T^{2}
23 1−6.39T+23T2 1 - 6.39T + 23T^{2}
29 1+(0.117+0.362i)T+(−23.4+17.0i)T2 1 + (0.117 + 0.362i)T + (-23.4 + 17.0i)T^{2}
31 1+(0.615+0.446i)T+(9.57+29.4i)T2 1 + (0.615 + 0.446i)T + (9.57 + 29.4i)T^{2}
37 1+(0.448+1.38i)T+(−29.9+21.7i)T2 1 + (0.448 + 1.38i)T + (-29.9 + 21.7i)T^{2}
41 1+(1.89−5.82i)T+(−33.1−24.0i)T2 1 + (1.89 - 5.82i)T + (-33.1 - 24.0i)T^{2}
43 1−7.19T+43T2 1 - 7.19T + 43T^{2}
47 1+(1.33−4.11i)T+(−38.0−27.6i)T2 1 + (1.33 - 4.11i)T + (-38.0 - 27.6i)T^{2}
53 1+(−5.62−4.08i)T+(16.3+50.4i)T2 1 + (-5.62 - 4.08i)T + (16.3 + 50.4i)T^{2}
59 1+(3.92+12.0i)T+(−47.7+34.6i)T2 1 + (3.92 + 12.0i)T + (-47.7 + 34.6i)T^{2}
61 1+(−7.19+5.22i)T+(18.8−58.0i)T2 1 + (-7.19 + 5.22i)T + (18.8 - 58.0i)T^{2}
67 1+12.5T+67T2 1 + 12.5T + 67T^{2}
71 1+(−5.95+4.32i)T+(21.9−67.5i)T2 1 + (-5.95 + 4.32i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.17−6.68i)T+(−59.0+42.9i)T2 1 + (-2.17 - 6.68i)T + (-59.0 + 42.9i)T^{2}
79 1+(5.65+4.11i)T+(24.4+75.1i)T2 1 + (5.65 + 4.11i)T + (24.4 + 75.1i)T^{2}
83 1+(8.69−6.31i)T+(25.6−78.9i)T2 1 + (8.69 - 6.31i)T + (25.6 - 78.9i)T^{2}
89 1−0.451T+89T2 1 - 0.451T + 89T^{2}
97 1+(−2.24−1.63i)T+(29.9+92.2i)T2 1 + (-2.24 - 1.63i)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

−9.482699068729691632864833316623, −8.713870906164138796936673428060, −8.061880529450913168861334394005, −7.10675176560281375652539384462, −6.49039733799253736833932208943, −5.88221784323415626807145619572, −4.03328802950578394513163961759, −3.14723107886132476039404816423, −1.68163082025527649219415883263, −0.894878432383776502689485322093, 2.43141283534897768680821204705, 3.12847978980293193779670340543, 4.20944091239351956889653283468, 5.15675559866730233402486488106, 5.86640939958215537847841280054, 7.06168429165983858311183871412, 8.632133525682987108524352040578, 8.920455489438658199613974206891, 9.531127453347271166687969724828, 10.41746572037621769775865563473

Graph of the ZZ-function along the critical line