↑

Properties

Label 2-880-11.3-c1-0-21
Degree 22
Conductor 880880
Sign −0.999−0.00395i-0.999 - 0.00395i
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  + (−0.655 − 2.01i)3-s + (0.809 + 0.587i)5-s + (−0.0946 + 0.291i)7-s + (−1.21 + 0.884i)9-s + (−2.72 − 1.89i)11-s + (−2.68 + 1.95i)13-s + (0.655 − 2.01i)15-s + (−4.58 − 3.33i)17-s + (−0.464 − 1.43i)19-s + 0.650·21-s + 0.343·23-s + (0.309 + 0.951i)25-s + (−2.56 − 1.86i)27-s + (2.15 − 6.64i)29-s + (−4.80 + 3.49i)31-s + ⋯
L(s)  = 1  + (−0.378 − 1.16i)3-s + (0.361 + 0.262i)5-s + (−0.0357 + 0.110i)7-s + (−0.405 + 0.294i)9-s + (−0.820 − 0.571i)11-s + (−0.745 + 0.541i)13-s + (0.169 − 0.521i)15-s + (−1.11 − 0.808i)17-s + (−0.106 − 0.328i)19-s + 0.141·21-s + 0.0716·23-s + (0.0618 + 0.190i)25-s + (−0.494 − 0.359i)27-s + (0.400 − 1.23i)29-s + (−0.863 + 0.627i)31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.999−0.00395i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00395i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=((−0.999−0.00395i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00395i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.999−0.00395i-0.999 - 0.00395i
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), −0.999−0.00395i)(2,\ 880,\ (\ :1/2),\ -0.999 - 0.00395i)

Particular Values

L(1)L(1) ≈\approx 0.00122742+0.619932i0.00122742 + 0.619932i
L(12)L(\frac12) ≈\approx 0.00122742+0.619932i0.00122742 + 0.619932i
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+(2.72+1.89i)T 1 + (2.72 + 1.89i)T
good3 1+(0.655+2.01i)T+(−2.42+1.76i)T2 1 + (0.655 + 2.01i)T + (-2.42 + 1.76i)T^{2}
7 1+(0.0946−0.291i)T+(−5.66−4.11i)T2 1 + (0.0946 - 0.291i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.68−1.95i)T+(4.01−12.3i)T2 1 + (2.68 - 1.95i)T + (4.01 - 12.3i)T^{2}
17 1+(4.58+3.33i)T+(5.25+16.1i)T2 1 + (4.58 + 3.33i)T + (5.25 + 16.1i)T^{2}
19 1+(0.464+1.43i)T+(−15.3+11.1i)T2 1 + (0.464 + 1.43i)T + (-15.3 + 11.1i)T^{2}
23 1−0.343T+23T2 1 - 0.343T + 23T^{2}
29 1+(−2.15+6.64i)T+(−23.4−17.0i)T2 1 + (-2.15 + 6.64i)T + (-23.4 - 17.0i)T^{2}
31 1+(4.80−3.49i)T+(9.57−29.4i)T2 1 + (4.80 - 3.49i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.63+5.04i)T+(−29.9−21.7i)T2 1 + (-1.63 + 5.04i)T + (-29.9 - 21.7i)T^{2}
41 1+(2.25+6.94i)T+(−33.1+24.0i)T2 1 + (2.25 + 6.94i)T + (-33.1 + 24.0i)T^{2}
43 1+4.16T+43T2 1 + 4.16T + 43T^{2}
47 1+(−1.94−5.98i)T+(−38.0+27.6i)T2 1 + (-1.94 - 5.98i)T + (-38.0 + 27.6i)T^{2}
53 1+(8.63−6.27i)T+(16.3−50.4i)T2 1 + (8.63 - 6.27i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.590+1.81i)T+(−47.7−34.6i)T2 1 + (-0.590 + 1.81i)T + (-47.7 - 34.6i)T^{2}
61 1+(8.27+6.01i)T+(18.8+58.0i)T2 1 + (8.27 + 6.01i)T + (18.8 + 58.0i)T^{2}
67 1+10.4T+67T2 1 + 10.4T + 67T^{2}
71 1+(−9.03−6.56i)T+(21.9+67.5i)T2 1 + (-9.03 - 6.56i)T + (21.9 + 67.5i)T^{2}
73 1+(0.792−2.43i)T+(−59.0−42.9i)T2 1 + (0.792 - 2.43i)T + (-59.0 - 42.9i)T^{2}
79 1+(−1.95+1.42i)T+(24.4−75.1i)T2 1 + (-1.95 + 1.42i)T + (24.4 - 75.1i)T^{2}
83 1+(−3.66−2.66i)T+(25.6+78.9i)T2 1 + (-3.66 - 2.66i)T + (25.6 + 78.9i)T^{2}
89 1−2.46T+89T2 1 - 2.46T + 89T^{2}
97 1+(11.1−8.06i)T+(29.9−92.2i)T2 1 + (11.1 - 8.06i)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

−9.647488936837829380543902266728, −8.887697218946953342946203976823, −7.74856209544487804931098401105, −7.11909870531731466384954114392, −6.37566230455774823185720357889, −5.53511989464766170654969055578, −4.45892606660809449956476477310, −2.80216734870516798793046712332, −1.95971175837046930014716861527, −0.28802360970746959906791391214, 2.00376000154456963888711008251, 3.38180649700657143863659282527, 4.59664344102306633169578775674, 5.02506210280135076867837148641, 6.01727575645440070828217315253, 7.14151938549471722883588905433, 8.153055751877517771621777281833, 9.093940753449519282460001403725, 9.936831157011614391650910052195, 10.41243538461290658481830767828

Graph of the ZZ-function along the critical line