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Properties

Label 2-220-11.6-c2-0-6
Degree 22
Conductor 220220
Sign −0.972+0.233i-0.972 + 0.233i
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  + (−2.30 − 1.67i)3-s + (0.690 − 2.12i)5-s + (1.86 + 2.56i)7-s + (−0.274 − 0.844i)9-s + (−8.20 − 7.32i)11-s + (1.09 − 0.355i)13-s + (−5.15 + 3.74i)15-s + (−16.6 − 5.41i)17-s + (−11.7 + 16.2i)19-s − 9.01i·21-s − 36.9·23-s + (−4.04 − 2.93i)25-s + (−8.70 + 26.7i)27-s + (−2.94 − 4.04i)29-s + (−18.8 − 58.1i)31-s + ⋯
L(s)  = 1  + (−0.768 − 0.558i)3-s + (0.138 − 0.425i)5-s + (0.265 + 0.365i)7-s + (−0.0304 − 0.0938i)9-s + (−0.746 − 0.665i)11-s + (0.0842 − 0.0273i)13-s + (−0.343 + 0.249i)15-s + (−0.980 − 0.318i)17-s + (−0.619 + 0.852i)19-s − 0.429i·21-s − 1.60·23-s + (−0.161 − 0.117i)25-s + (−0.322 + 0.992i)27-s + (−0.101 − 0.139i)29-s + (−0.609 − 1.87i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.972+0.233i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.972+0.233i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.972+0.233i-0.972 + 0.233i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(61,⋅)\chi_{220} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.972+0.233i)(2,\ 220,\ (\ :1),\ -0.972 + 0.233i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0616145−0.519665i0.0616145 - 0.519665i
L(12)L(\frac12) ≈\approx 0.0616145−0.519665i0.0616145 - 0.519665i
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+(−0.690+2.12i)T 1 + (-0.690 + 2.12i)T
11 1+(8.20+7.32i)T 1 + (8.20 + 7.32i)T
good3 1+(2.30+1.67i)T+(2.78+8.55i)T2 1 + (2.30 + 1.67i)T + (2.78 + 8.55i)T^{2}
7 1+(−1.86−2.56i)T+(−15.1+46.6i)T2 1 + (-1.86 - 2.56i)T + (-15.1 + 46.6i)T^{2}
13 1+(−1.09+0.355i)T+(136.−99.3i)T2 1 + (-1.09 + 0.355i)T + (136. - 99.3i)T^{2}
17 1+(16.6+5.41i)T+(233.+169.i)T2 1 + (16.6 + 5.41i)T + (233. + 169. i)T^{2}
19 1+(11.7−16.2i)T+(−111.−343.i)T2 1 + (11.7 - 16.2i)T + (-111. - 343. i)T^{2}
23 1+36.9T+529T2 1 + 36.9T + 529T^{2}
29 1+(2.94+4.04i)T+(−259.+799.i)T2 1 + (2.94 + 4.04i)T + (-259. + 799. i)T^{2}
31 1+(18.8+58.1i)T+(−777.+564.i)T2 1 + (18.8 + 58.1i)T + (-777. + 564. i)T^{2}
37 1+(−21.0+15.3i)T+(423.−1.30e3i)T2 1 + (-21.0 + 15.3i)T + (423. - 1.30e3i)T^{2}
41 1+(−35.4+48.7i)T+(−519.−1.59e3i)T2 1 + (-35.4 + 48.7i)T + (-519. - 1.59e3i)T^{2}
43 1−65.3iT−1.84e3T2 1 - 65.3iT - 1.84e3T^{2}
47 1+(−21.5−15.6i)T+(682.+2.10e3i)T2 1 + (-21.5 - 15.6i)T + (682. + 2.10e3i)T^{2}
53 1+(−0.957−2.94i)T+(−2.27e3+1.65e3i)T2 1 + (-0.957 - 2.94i)T + (-2.27e3 + 1.65e3i)T^{2}
59 1+(−7.53+5.47i)T+(1.07e3−3.31e3i)T2 1 + (-7.53 + 5.47i)T + (1.07e3 - 3.31e3i)T^{2}
61 1+(−32.2−10.4i)T+(3.01e3+2.18e3i)T2 1 + (-32.2 - 10.4i)T + (3.01e3 + 2.18e3i)T^{2}
67 1+17.1T+4.48e3T2 1 + 17.1T + 4.48e3T^{2}
71 1+(−15.6+48.2i)T+(−4.07e3−2.96e3i)T2 1 + (-15.6 + 48.2i)T + (-4.07e3 - 2.96e3i)T^{2}
73 1+(−24.2−33.4i)T+(−1.64e3+5.06e3i)T2 1 + (-24.2 - 33.4i)T + (-1.64e3 + 5.06e3i)T^{2}
79 1+(−40.1+13.0i)T+(5.04e3−3.66e3i)T2 1 + (-40.1 + 13.0i)T + (5.04e3 - 3.66e3i)T^{2}
83 1+(76.7+24.9i)T+(5.57e3+4.04e3i)T2 1 + (76.7 + 24.9i)T + (5.57e3 + 4.04e3i)T^{2}
89 1+109.T+7.92e3T2 1 + 109.T + 7.92e3T^{2}
97 1+(6.19+19.0i)T+(−7.61e3+5.53e3i)T2 1 + (6.19 + 19.0i)T + (-7.61e3 + 5.53e3i)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

−11.62124275099520265952246326054, −10.93537070157766438737122491418, −9.656614806556662843234681001697, −8.540174766216934750981280464186, −7.60037417085496421990951326675, −6.12498985584016253501535750507, −5.65967798692295421345379777659, −4.13340929230807037100050949784, −2.15667818007895407230486585987, −0.29613196059318324173255410984, 2.28383000673392033381122036026, 4.17514954444824312394322263785, 5.09618773095087372106309340497, 6.28291571252557014546467197414, 7.38746790092267431604493507460, 8.573204511407609116166596817754, 9.951156059085538697223626067120, 10.65269805486922043775384452168, 11.23555019603588665187453646331, 12.40408941377872597952216023296

Graph of the ZZ-function along the critical line