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Properties

Label 2-220-44.7-c1-0-10
Degree 22
Conductor 220220
Sign 0.446−0.894i0.446 - 0.894i
Analytic cond. 1.756701.75670
Root an. cond. 1.325401.32540
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  + (1.25 + 0.659i)2-s + (0.260 + 0.0846i)3-s + (1.13 + 1.64i)4-s + (−0.809 + 0.587i)5-s + (0.269 + 0.277i)6-s + (0.560 + 1.72i)7-s + (0.326 + 2.80i)8-s + (−2.36 − 1.71i)9-s + (−1.39 + 0.201i)10-s + (2.57 + 2.08i)11-s + (0.154 + 0.525i)12-s + (2.98 − 4.10i)13-s + (−0.436 + 2.52i)14-s + (−0.260 + 0.0846i)15-s + (−1.44 + 3.73i)16-s + (−2.63 − 3.62i)17-s + ⋯
L(s)  = 1  + (0.884 + 0.466i)2-s + (0.150 + 0.0488i)3-s + (0.565 + 0.824i)4-s + (−0.361 + 0.262i)5-s + (0.110 + 0.113i)6-s + (0.211 + 0.651i)7-s + (0.115 + 0.993i)8-s + (−0.788 − 0.573i)9-s + (−0.442 + 0.0638i)10-s + (0.777 + 0.628i)11-s + (0.0446 + 0.151i)12-s + (0.826 − 1.13i)13-s + (−0.116 + 0.675i)14-s + (−0.0672 + 0.0218i)15-s + (−0.361 + 0.932i)16-s + (−0.638 − 0.878i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.446−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.446−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.446−0.894i0.446 - 0.894i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(51,⋅)\chi_{220} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.446−0.894i)(2,\ 220,\ (\ :1/2),\ 0.446 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.65939+1.02688i1.65939 + 1.02688i
L(12)L(\frac12) ≈\approx 1.65939+1.02688i1.65939 + 1.02688i
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+(−1.25−0.659i)T 1 + (-1.25 - 0.659i)T
5 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
11 1+(−2.57−2.08i)T 1 + (-2.57 - 2.08i)T
good3 1+(−0.260−0.0846i)T+(2.42+1.76i)T2 1 + (-0.260 - 0.0846i)T + (2.42 + 1.76i)T^{2}
7 1+(−0.560−1.72i)T+(−5.66+4.11i)T2 1 + (-0.560 - 1.72i)T + (-5.66 + 4.11i)T^{2}
13 1+(−2.98+4.10i)T+(−4.01−12.3i)T2 1 + (-2.98 + 4.10i)T + (-4.01 - 12.3i)T^{2}
17 1+(2.63+3.62i)T+(−5.25+16.1i)T2 1 + (2.63 + 3.62i)T + (-5.25 + 16.1i)T^{2}
19 1+(−0.882+2.71i)T+(−15.3−11.1i)T2 1 + (-0.882 + 2.71i)T + (-15.3 - 11.1i)T^{2}
23 1+0.353iT−23T2 1 + 0.353iT - 23T^{2}
29 1+(3.78−1.23i)T+(23.4−17.0i)T2 1 + (3.78 - 1.23i)T + (23.4 - 17.0i)T^{2}
31 1+(−0.598+0.823i)T+(−9.57−29.4i)T2 1 + (-0.598 + 0.823i)T + (-9.57 - 29.4i)T^{2}
37 1+(1.34+4.14i)T+(−29.9+21.7i)T2 1 + (1.34 + 4.14i)T + (-29.9 + 21.7i)T^{2}
41 1+(−3.62−1.17i)T+(33.1+24.0i)T2 1 + (-3.62 - 1.17i)T + (33.1 + 24.0i)T^{2}
43 1+11.9T+43T2 1 + 11.9T + 43T^{2}
47 1+(−10.2−3.33i)T+(38.0+27.6i)T2 1 + (-10.2 - 3.33i)T + (38.0 + 27.6i)T^{2}
53 1+(4.99+3.62i)T+(16.3+50.4i)T2 1 + (4.99 + 3.62i)T + (16.3 + 50.4i)T^{2}
59 1+(9.49−3.08i)T+(47.7−34.6i)T2 1 + (9.49 - 3.08i)T + (47.7 - 34.6i)T^{2}
61 1+(−7.97−10.9i)T+(−18.8+58.0i)T2 1 + (-7.97 - 10.9i)T + (-18.8 + 58.0i)T^{2}
67 1+12.6iT−67T2 1 + 12.6iT - 67T^{2}
71 1+(−1.91−2.63i)T+(−21.9+67.5i)T2 1 + (-1.91 - 2.63i)T + (-21.9 + 67.5i)T^{2}
73 1+(1.42−0.463i)T+(59.0−42.9i)T2 1 + (1.42 - 0.463i)T + (59.0 - 42.9i)T^{2}
79 1+(−10.3−7.53i)T+(24.4+75.1i)T2 1 + (-10.3 - 7.53i)T + (24.4 + 75.1i)T^{2}
83 1+(8.34−6.06i)T+(25.6−78.9i)T2 1 + (8.34 - 6.06i)T + (25.6 - 78.9i)T^{2}
89 1+1.53T+89T2 1 + 1.53T + 89T^{2}
97 1+(6.18+4.49i)T+(29.9+92.2i)T2 1 + (6.18 + 4.49i)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

−12.44577980052329479530836231802, −11.66985993888688313984734181253, −10.96762955792088112208003628844, −9.232018292442012515169845979086, −8.404207001207297295923907706367, −7.24291256864793866757943970797, −6.21086329562625137044824442421, −5.18522500088263977453871355809, −3.79913737038140130758222057957, −2.67660069987443737981825012533, 1.66012849052636055620186476570, 3.53070058255571875370886638314, 4.36692741522658801330528594410, 5.78203634051181266189329167156, 6.77314819783482800974989278855, 8.179447643466625187645474077777, 9.187454769551471860805920017183, 10.63320319370674703827999592745, 11.28832280858181430976383323980, 11.98368925734759327139776405396

Graph of the ZZ-function along the critical line