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Properties

Label 2-880-55.32-c1-0-19
Degree 22
Conductor 880880
Sign 0.978+0.207i0.978 + 0.207i
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  + (−1.70 + 1.70i)3-s + (−0.707 + 2.12i)5-s + (2.12 − 2.12i)7-s − 2.82i·9-s + (1.41 − 3i)11-s + (−3 − 3i)13-s + (−2.41 − 4.82i)15-s + (5.12 − 5.12i)17-s − 3·19-s + 7.24i·21-s + (0.171 − 0.171i)23-s + (−3.99 − 3i)25-s + (−0.292 − 0.292i)27-s + 1.24·29-s + 7.24·31-s + ⋯
L(s)  = 1  + (−0.985 + 0.985i)3-s + (−0.316 + 0.948i)5-s + (0.801 − 0.801i)7-s − 0.942i·9-s + (0.426 − 0.904i)11-s + (−0.832 − 0.832i)13-s + (−0.623 − 1.24i)15-s + (1.24 − 1.24i)17-s − 0.688·19-s + 1.58i·21-s + (0.0357 − 0.0357i)23-s + (−0.799 − 0.600i)25-s + (−0.0563 − 0.0563i)27-s + 0.230·29-s + 1.30·31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((0.978+0.207i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((0.978+0.207i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: 0.978+0.207i0.978 + 0.207i
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ880(417,⋅)\chi_{880} (417, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 880, ( :1/2), 0.978+0.207i)(2,\ 880,\ (\ :1/2),\ 0.978 + 0.207i)

Particular Values

L(1)L(1) ≈\approx 0.979511−0.102578i0.979511 - 0.102578i
L(12)L(\frac12) ≈\approx 0.979511−0.102578i0.979511 - 0.102578i
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+(0.707−2.12i)T 1 + (0.707 - 2.12i)T
11 1+(−1.41+3i)T 1 + (-1.41 + 3i)T
good3 1+(1.70−1.70i)T−3iT2 1 + (1.70 - 1.70i)T - 3iT^{2}
7 1+(−2.12+2.12i)T−7iT2 1 + (-2.12 + 2.12i)T - 7iT^{2}
13 1+(3+3i)T+13iT2 1 + (3 + 3i)T + 13iT^{2}
17 1+(−5.12+5.12i)T−17iT2 1 + (-5.12 + 5.12i)T - 17iT^{2}
19 1+3T+19T2 1 + 3T + 19T^{2}
23 1+(−0.171+0.171i)T−23iT2 1 + (-0.171 + 0.171i)T - 23iT^{2}
29 1−1.24T+29T2 1 - 1.24T + 29T^{2}
31 1−7.24T+31T2 1 - 7.24T + 31T^{2}
37 1+(−0.121−0.121i)T+37iT2 1 + (-0.121 - 0.121i)T + 37iT^{2}
41 1−1.75iT−41T2 1 - 1.75iT - 41T^{2}
43 1+(−1.24−1.24i)T+43iT2 1 + (-1.24 - 1.24i)T + 43iT^{2}
47 1+(4.41+4.41i)T+47iT2 1 + (4.41 + 4.41i)T + 47iT^{2}
53 1+(−9.53+9.53i)T−53iT2 1 + (-9.53 + 9.53i)T - 53iT^{2}
59 1+1.41iT−59T2 1 + 1.41iT - 59T^{2}
61 1−7.24iT−61T2 1 - 7.24iT - 61T^{2}
67 1+(−4−4i)T+67iT2 1 + (-4 - 4i)T + 67iT^{2}
71 1−1.24T+71T2 1 - 1.24T + 71T^{2}
73 1+(6+6i)T+73iT2 1 + (6 + 6i)T + 73iT^{2}
79 1−10.2T+79T2 1 - 10.2T + 79T^{2}
83 1+(7.24+7.24i)T+83iT2 1 + (7.24 + 7.24i)T + 83iT^{2}
89 1−5.48iT−89T2 1 - 5.48iT - 89T^{2}
97 1+(2.24+2.24i)T+97iT2 1 + (2.24 + 2.24i)T + 97iT^{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

−10.28579926623106580827191754703, −9.739427736940178006268992287345, −8.281741812066952630818560613144, −7.54194834985726014603808613973, −6.60204485530820920592888676283, −5.58055815458882746690506514246, −4.81062179527218342776315664922, −3.88836829419800040627736164487, −2.85678901668547518258239522610, −0.62036459299959450696983813829, 1.24991673074454479787185590456, 2.08347937076167921290407390038, 4.17178525684872472893920718355, 4.99417860622411293642445655385, 5.79639053376877714650266302284, 6.67724260205588079154329630437, 7.64086308166406628007243959183, 8.314864136182213212673839239709, 9.237136504161169172891030396134, 10.20600342620145185713160874130

Graph of the ZZ-function along the critical line