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Properties

Label 2-220-4.3-c2-0-4
Degree 22
Conductor 220220
Sign −0.272−0.962i-0.272 - 0.962i
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  + (−0.274 − 1.98i)2-s + 4.52i·3-s + (−3.84 + 1.08i)4-s + 2.23·5-s + (8.96 − 1.24i)6-s + 1.58i·7-s + (3.21 + 7.32i)8-s − 11.5·9-s + (−0.614 − 4.42i)10-s + 3.31i·11-s + (−4.92 − 17.4i)12-s − 18.2·13-s + (3.13 − 0.435i)14-s + 10.1i·15-s + (13.6 − 8.37i)16-s − 13.8·17-s + ⋯
L(s)  = 1  + (−0.137 − 0.990i)2-s + 1.50i·3-s + (−0.962 + 0.272i)4-s + 0.447·5-s + (1.49 − 0.207i)6-s + 0.226i·7-s + (0.401 + 0.915i)8-s − 1.27·9-s + (−0.0614 − 0.442i)10-s + 0.301i·11-s + (−0.410 − 1.45i)12-s − 1.40·13-s + (0.224 − 0.0310i)14-s + 0.674i·15-s + (0.851 − 0.523i)16-s − 0.812·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.272−0.962i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.272−0.962i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.272−0.962i-0.272 - 0.962i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(111,⋅)\chi_{220} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.272−0.962i)(2,\ 220,\ (\ :1),\ -0.272 - 0.962i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.560355+0.740728i0.560355 + 0.740728i
L(12)L(\frac12) ≈\approx 0.560355+0.740728i0.560355 + 0.740728i
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+(0.274+1.98i)T 1 + (0.274 + 1.98i)T
5 1−2.23T 1 - 2.23T
11 1−3.31iT 1 - 3.31iT
good3 1−4.52iT−9T2 1 - 4.52iT - 9T^{2}
7 1−1.58iT−49T2 1 - 1.58iT - 49T^{2}
13 1+18.2T+169T2 1 + 18.2T + 169T^{2}
17 1+13.8T+289T2 1 + 13.8T + 289T^{2}
19 1−13.5iT−361T2 1 - 13.5iT - 361T^{2}
23 1−34.1iT−529T2 1 - 34.1iT - 529T^{2}
29 1+26.1T+841T2 1 + 26.1T + 841T^{2}
31 1+0.621iT−961T2 1 + 0.621iT - 961T^{2}
37 1−9.06T+1.36e3T2 1 - 9.06T + 1.36e3T^{2}
41 1−42.3T+1.68e3T2 1 - 42.3T + 1.68e3T^{2}
43 1+30.1iT−1.84e3T2 1 + 30.1iT - 1.84e3T^{2}
47 1−46.6iT−2.20e3T2 1 - 46.6iT - 2.20e3T^{2}
53 1−23.6T+2.80e3T2 1 - 23.6T + 2.80e3T^{2}
59 1+88.7iT−3.48e3T2 1 + 88.7iT - 3.48e3T^{2}
61 1+49.4T+3.72e3T2 1 + 49.4T + 3.72e3T^{2}
67 1−113.iT−4.48e3T2 1 - 113. iT - 4.48e3T^{2}
71 1+55.1iT−5.04e3T2 1 + 55.1iT - 5.04e3T^{2}
73 1−123.T+5.32e3T2 1 - 123.T + 5.32e3T^{2}
79 1+0.962iT−6.24e3T2 1 + 0.962iT - 6.24e3T^{2}
83 1+29.6iT−6.88e3T2 1 + 29.6iT - 6.88e3T^{2}
89 1−57.8T+7.92e3T2 1 - 57.8T + 7.92e3T^{2}
97 1−176.T+9.40e3T2 1 - 176.T + 9.40e3T^{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

−12.10593033707531486751808310456, −11.15482660305551638630050681352, −10.30852787813709582729028037547, −9.536173041467879644053923389518, −9.118909497586796520655673450335, −7.62740354676652230694999969494, −5.57042785822574050414123430937, −4.70381782630483524903859654710, −3.64897944556010362159131349432, −2.25465511172402584606429050934, 0.52159366675445576672225233581, 2.34525410928129014421702066457, 4.63512502307635427719142864183, 5.93046829357846517725596740504, 6.84664042393251876099231655440, 7.45313528173887018331174366682, 8.511609903474244077840652835264, 9.485655237450933201638279510181, 10.76046335357918835823065072114, 12.19443227425325026789536697200

Graph of the ZZ-function along the critical line