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Properties

Label 2-220-44.7-c1-0-6
Degree 22
Conductor 220220
Sign 0.233+0.972i0.233 + 0.972i
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  + (−1.26 − 0.627i)2-s + (−1.88 − 0.612i)3-s + (1.21 + 1.59i)4-s + (0.809 − 0.587i)5-s + (2.00 + 1.96i)6-s + (0.668 + 2.05i)7-s + (−0.539 − 2.77i)8-s + (0.755 + 0.548i)9-s + (−1.39 + 0.237i)10-s + (3.31 + 0.105i)11-s + (−1.31 − 3.74i)12-s + (1.85 − 2.55i)13-s + (0.443 − 3.02i)14-s + (−1.88 + 0.612i)15-s + (−1.05 + 3.85i)16-s + (−2.62 − 3.60i)17-s + ⋯
L(s)  = 1  + (−0.896 − 0.443i)2-s + (−1.08 − 0.353i)3-s + (0.606 + 0.795i)4-s + (0.361 − 0.262i)5-s + (0.819 + 0.800i)6-s + (0.252 + 0.777i)7-s + (−0.190 − 0.981i)8-s + (0.251 + 0.182i)9-s + (−0.440 + 0.0750i)10-s + (0.999 + 0.0316i)11-s + (−0.379 − 1.08i)12-s + (0.515 − 0.709i)13-s + (0.118 − 0.809i)14-s + (−0.487 + 0.158i)15-s + (−0.264 + 0.964i)16-s + (−0.635 − 0.874i)17-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.233+0.972i0.233 + 0.972i
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), 0.233+0.972i)(2,\ 220,\ (\ :1/2),\ 0.233 + 0.972i)

Particular Values

L(1)L(1) ≈\approx 0.488553−0.384918i0.488553 - 0.384918i
L(12)L(\frac12) ≈\approx 0.488553−0.384918i0.488553 - 0.384918i
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+(1.26+0.627i)T 1 + (1.26 + 0.627i)T
5 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
11 1+(−3.31−0.105i)T 1 + (-3.31 - 0.105i)T
good3 1+(1.88+0.612i)T+(2.42+1.76i)T2 1 + (1.88 + 0.612i)T + (2.42 + 1.76i)T^{2}
7 1+(−0.668−2.05i)T+(−5.66+4.11i)T2 1 + (-0.668 - 2.05i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.85+2.55i)T+(−4.01−12.3i)T2 1 + (-1.85 + 2.55i)T + (-4.01 - 12.3i)T^{2}
17 1+(2.62+3.60i)T+(−5.25+16.1i)T2 1 + (2.62 + 3.60i)T + (-5.25 + 16.1i)T^{2}
19 1+(−1.21+3.74i)T+(−15.3−11.1i)T2 1 + (-1.21 + 3.74i)T + (-15.3 - 11.1i)T^{2}
23 1+8.67iT−23T2 1 + 8.67iT - 23T^{2}
29 1+(−6.12+1.99i)T+(23.4−17.0i)T2 1 + (-6.12 + 1.99i)T + (23.4 - 17.0i)T^{2}
31 1+(0.673−0.927i)T+(−9.57−29.4i)T2 1 + (0.673 - 0.927i)T + (-9.57 - 29.4i)T^{2}
37 1+(−3.17−9.78i)T+(−29.9+21.7i)T2 1 + (-3.17 - 9.78i)T + (-29.9 + 21.7i)T^{2}
41 1+(1.29+0.421i)T+(33.1+24.0i)T2 1 + (1.29 + 0.421i)T + (33.1 + 24.0i)T^{2}
43 1+4.00T+43T2 1 + 4.00T + 43T^{2}
47 1+(−3.61−1.17i)T+(38.0+27.6i)T2 1 + (-3.61 - 1.17i)T + (38.0 + 27.6i)T^{2}
53 1+(−6.21−4.51i)T+(16.3+50.4i)T2 1 + (-6.21 - 4.51i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.112+0.0366i)T+(47.7−34.6i)T2 1 + (-0.112 + 0.0366i)T + (47.7 - 34.6i)T^{2}
61 1+(7.07+9.74i)T+(−18.8+58.0i)T2 1 + (7.07 + 9.74i)T + (-18.8 + 58.0i)T^{2}
67 1−6.69iT−67T2 1 - 6.69iT - 67T^{2}
71 1+(0.556+0.766i)T+(−21.9+67.5i)T2 1 + (0.556 + 0.766i)T + (-21.9 + 67.5i)T^{2}
73 1+(−1.45+0.471i)T+(59.0−42.9i)T2 1 + (-1.45 + 0.471i)T + (59.0 - 42.9i)T^{2}
79 1+(3.49+2.53i)T+(24.4+75.1i)T2 1 + (3.49 + 2.53i)T + (24.4 + 75.1i)T^{2}
83 1+(−0.790+0.574i)T+(25.6−78.9i)T2 1 + (-0.790 + 0.574i)T + (25.6 - 78.9i)T^{2}
89 1+13.2T+89T2 1 + 13.2T + 89T^{2}
97 1+(−11.7−8.52i)T+(29.9+92.2i)T2 1 + (-11.7 - 8.52i)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

−11.89010530603572360982992662835, −11.25176303146210697988742531117, −10.26914629483125830528139178925, −9.066670993631965027493072274439, −8.436078263932833474915328917093, −6.83115657455616305474415272960, −6.17761854652206348656129562925, −4.76960222343025091658111338697, −2.69756038019811534428764253522, −0.907462102749955900493532762874, 1.47679682169807218339312834579, 4.08934291552501013696870829348, 5.59633662589896588083797546851, 6.36305735165699400323632235048, 7.31666539140329952540481791194, 8.644926225352095163268942746399, 9.696558419705093250784650684264, 10.58996349938243587182017056071, 11.22133367057719886449893381888, 11.98134171073134742591159952258

Graph of the ZZ-function along the critical line