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Properties

Label 2-220-11.7-c2-0-7
Degree 22
Conductor 220220
Sign −0.997+0.0682i-0.997 + 0.0682i
Analytic cond. 5.994565.99456
Root an. cond. 2.448382.44838
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.59 − 4.90i)3-s + (−1.80 + 1.31i)5-s + (−9.66 + 3.13i)7-s + (−14.2 − 10.3i)9-s + (−1.51 − 10.8i)11-s + (−1.96 + 2.70i)13-s + (3.56 + 10.9i)15-s + (−11.8 − 16.3i)17-s + (−5.26 − 1.70i)19-s + 52.3i·21-s + 34.6·23-s + (1.54 − 4.75i)25-s + (−35.9 + 26.1i)27-s + (−42.6 + 13.8i)29-s + (35.0 + 25.4i)31-s + ⋯
L(s)  = 1  + (0.531 − 1.63i)3-s + (−0.361 + 0.262i)5-s + (−1.38 + 0.448i)7-s + (−1.58 − 1.15i)9-s + (−0.137 − 0.990i)11-s + (−0.151 + 0.208i)13-s + (0.237 + 0.731i)15-s + (−0.697 − 0.960i)17-s + (−0.276 − 0.0899i)19-s + 2.49i·21-s + 1.50·23-s + (0.0618 − 0.190i)25-s + (−1.33 + 0.967i)27-s + (−1.47 + 0.478i)29-s + (1.13 + 0.821i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.997+0.0682i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0682i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.997+0.0682i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.997+0.0682i-0.997 + 0.0682i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(161,⋅)\chi_{220} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.997+0.0682i)(2,\ 220,\ (\ :1),\ -0.997 + 0.0682i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0307434−0.900053i0.0307434 - 0.900053i
L(12)L(\frac12) ≈\approx 0.0307434−0.900053i0.0307434 - 0.900053i
L(2)L(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+(1.80−1.31i)T 1 + (1.80 - 1.31i)T
11 1+(1.51+10.8i)T 1 + (1.51 + 10.8i)T
good3 1+(−1.59+4.90i)T+(−7.28−5.29i)T2 1 + (-1.59 + 4.90i)T + (-7.28 - 5.29i)T^{2}
7 1+(9.66−3.13i)T+(39.6−28.8i)T2 1 + (9.66 - 3.13i)T + (39.6 - 28.8i)T^{2}
13 1+(1.96−2.70i)T+(−52.2−160.i)T2 1 + (1.96 - 2.70i)T + (-52.2 - 160. i)T^{2}
17 1+(11.8+16.3i)T+(−89.3+274.i)T2 1 + (11.8 + 16.3i)T + (-89.3 + 274. i)T^{2}
19 1+(5.26+1.70i)T+(292.+212.i)T2 1 + (5.26 + 1.70i)T + (292. + 212. i)T^{2}
23 1−34.6T+529T2 1 - 34.6T + 529T^{2}
29 1+(42.6−13.8i)T+(680.−494.i)T2 1 + (42.6 - 13.8i)T + (680. - 494. i)T^{2}
31 1+(−35.0−25.4i)T+(296.+913.i)T2 1 + (-35.0 - 25.4i)T + (296. + 913. i)T^{2}
37 1+(18.3+56.4i)T+(−1.10e3+804.i)T2 1 + (18.3 + 56.4i)T + (-1.10e3 + 804. i)T^{2}
41 1+(−31.4−10.2i)T+(1.35e3+988.i)T2 1 + (-31.4 - 10.2i)T + (1.35e3 + 988. i)T^{2}
43 1+65.3iT−1.84e3T2 1 + 65.3iT - 1.84e3T^{2}
47 1+(−21.2+65.4i)T+(−1.78e3−1.29e3i)T2 1 + (-21.2 + 65.4i)T + (-1.78e3 - 1.29e3i)T^{2}
53 1+(−2.60−1.89i)T+(868.+2.67e3i)T2 1 + (-2.60 - 1.89i)T + (868. + 2.67e3i)T^{2}
59 1+(−20.4−62.7i)T+(−2.81e3+2.04e3i)T2 1 + (-20.4 - 62.7i)T + (-2.81e3 + 2.04e3i)T^{2}
61 1+(27.8+38.3i)T+(−1.14e3+3.53e3i)T2 1 + (27.8 + 38.3i)T + (-1.14e3 + 3.53e3i)T^{2}
67 1+31.6T+4.48e3T2 1 + 31.6T + 4.48e3T^{2}
71 1+(3.04−2.21i)T+(1.55e3−4.79e3i)T2 1 + (3.04 - 2.21i)T + (1.55e3 - 4.79e3i)T^{2}
73 1+(−21.3+6.92i)T+(4.31e3−3.13e3i)T2 1 + (-21.3 + 6.92i)T + (4.31e3 - 3.13e3i)T^{2}
79 1+(45.9−63.2i)T+(−1.92e3−5.93e3i)T2 1 + (45.9 - 63.2i)T + (-1.92e3 - 5.93e3i)T^{2}
83 1+(−13.0−17.9i)T+(−2.12e3+6.55e3i)T2 1 + (-13.0 - 17.9i)T + (-2.12e3 + 6.55e3i)T^{2}
89 1−68.2T+7.92e3T2 1 - 68.2T + 7.92e3T^{2}
97 1+(49.7+36.1i)T+(2.90e3+8.94e3i)T2 1 + (49.7 + 36.1i)T + (2.90e3 + 8.94e3i)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

−11.87607019457032770446411850609, −10.84876907416488868999299067205, −9.177000852099654230605705394741, −8.658788354823533217882761788259, −7.23201988598608559329256400971, −6.81049906599984980476043385554, −5.67826082952489845059350479843, −3.36172093025384440815976477627, −2.47668865785485053276894684458, −0.44189440520618971610213197338, 2.89026941993046505290419078596, 3.97868860692315592050671687362, 4.76226564840442262394003692172, 6.31405260823126374577265091212, 7.71601481030485966094342774214, 8.964170056922250568273295013790, 9.656074836148478003387740469409, 10.32865402230244894089653105657, 11.24986825301565995443924783381, 12.73756668243342909435661654800

Graph of the ZZ-function along the critical line