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Properties

Label 2-880-11.3-c1-0-14
Degree 22
Conductor 880880
Sign 0.594+0.803i0.594 + 0.803i
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  + (−0.220 − 0.678i)3-s + (−0.809 − 0.587i)5-s + (−0.116 + 0.357i)7-s + (2.01 − 1.46i)9-s + (0.107 + 3.31i)11-s + (2.28 − 1.66i)13-s + (−0.220 + 0.678i)15-s + (3.91 + 2.84i)17-s + (−0.905 − 2.78i)19-s + 0.268·21-s − 3.77·23-s + (0.309 + 0.951i)25-s + (−3.16 − 2.30i)27-s + (2.60 − 8.03i)29-s + (6.50 − 4.72i)31-s + ⋯
L(s)  = 1  + (−0.127 − 0.391i)3-s + (−0.361 − 0.262i)5-s + (−0.0439 + 0.135i)7-s + (0.671 − 0.488i)9-s + (0.0322 + 0.999i)11-s + (0.634 − 0.461i)13-s + (−0.0569 + 0.175i)15-s + (0.949 + 0.689i)17-s + (−0.207 − 0.639i)19-s + 0.0585·21-s − 0.786·23-s + (0.0618 + 0.190i)25-s + (−0.609 − 0.443i)27-s + (0.484 − 1.49i)29-s + (1.16 − 0.848i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((0.594+0.803i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((0.594+0.803i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: 0.594+0.803i0.594 + 0.803i
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), 0.594+0.803i)(2,\ 880,\ (\ :1/2),\ 0.594 + 0.803i)

Particular Values

L(1)L(1) ≈\approx 1.33136−0.671236i1.33136 - 0.671236i
L(12)L(\frac12) ≈\approx 1.33136−0.671236i1.33136 - 0.671236i
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+(−0.107−3.31i)T 1 + (-0.107 - 3.31i)T
good3 1+(0.220+0.678i)T+(−2.42+1.76i)T2 1 + (0.220 + 0.678i)T + (-2.42 + 1.76i)T^{2}
7 1+(0.116−0.357i)T+(−5.66−4.11i)T2 1 + (0.116 - 0.357i)T + (-5.66 - 4.11i)T^{2}
13 1+(−2.28+1.66i)T+(4.01−12.3i)T2 1 + (-2.28 + 1.66i)T + (4.01 - 12.3i)T^{2}
17 1+(−3.91−2.84i)T+(5.25+16.1i)T2 1 + (-3.91 - 2.84i)T + (5.25 + 16.1i)T^{2}
19 1+(0.905+2.78i)T+(−15.3+11.1i)T2 1 + (0.905 + 2.78i)T + (-15.3 + 11.1i)T^{2}
23 1+3.77T+23T2 1 + 3.77T + 23T^{2}
29 1+(−2.60+8.03i)T+(−23.4−17.0i)T2 1 + (-2.60 + 8.03i)T + (-23.4 - 17.0i)T^{2}
31 1+(−6.50+4.72i)T+(9.57−29.4i)T2 1 + (-6.50 + 4.72i)T + (9.57 - 29.4i)T^{2}
37 1+(−0.877+2.70i)T+(−29.9−21.7i)T2 1 + (-0.877 + 2.70i)T + (-29.9 - 21.7i)T^{2}
41 1+(1.14+3.53i)T+(−33.1+24.0i)T2 1 + (1.14 + 3.53i)T + (-33.1 + 24.0i)T^{2}
43 1−6.48T+43T2 1 - 6.48T + 43T^{2}
47 1+(0.800+2.46i)T+(−38.0+27.6i)T2 1 + (0.800 + 2.46i)T + (-38.0 + 27.6i)T^{2}
53 1+(−0.0394+0.0286i)T+(16.3−50.4i)T2 1 + (-0.0394 + 0.0286i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.509+1.56i)T+(−47.7−34.6i)T2 1 + (-0.509 + 1.56i)T + (-47.7 - 34.6i)T^{2}
61 1+(7.03+5.11i)T+(18.8+58.0i)T2 1 + (7.03 + 5.11i)T + (18.8 + 58.0i)T^{2}
67 1+11.4T+67T2 1 + 11.4T + 67T^{2}
71 1+(−11.4−8.30i)T+(21.9+67.5i)T2 1 + (-11.4 - 8.30i)T + (21.9 + 67.5i)T^{2}
73 1+(−0.158+0.488i)T+(−59.0−42.9i)T2 1 + (-0.158 + 0.488i)T + (-59.0 - 42.9i)T^{2}
79 1+(−10.5+7.63i)T+(24.4−75.1i)T2 1 + (-10.5 + 7.63i)T + (24.4 - 75.1i)T^{2}
83 1+(−2.21−1.60i)T+(25.6+78.9i)T2 1 + (-2.21 - 1.60i)T + (25.6 + 78.9i)T^{2}
89 1−12.0T+89T2 1 - 12.0T + 89T^{2}
97 1+(13.0−9.44i)T+(29.9−92.2i)T2 1 + (13.0 - 9.44i)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

−9.971029824722937668986533115645, −9.276178089783979870466304099560, −8.092113585684683918531640118580, −7.61468902148330493545822462315, −6.52716256099334092730364907081, −5.81805018001164054838183064518, −4.49988318694834729374633426678, −3.78649485366392744960321214371, −2.26146453554270525046484533887, −0.883685741068940286410924034009, 1.28745732477701713372980737879, 3.01720645596976070288246653134, 3.90698911433499621269502536186, 4.87890066365125238839876061510, 5.91487655173309201527669676728, 6.83719958060996317364170119831, 7.79745009714599433228856908265, 8.517260687939952552868749426071, 9.530025526667447311884818070782, 10.43195351057346627961173933767

Graph of the ZZ-function along the critical line