↑

Properties

Label 2-880-11.4-c1-0-6
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

Related objects

Downloads

Learn more

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(81,⋅)\chi_{880} (81, \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}
show more
show less
   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

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

Graph of the ZZ-function along the critical line