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Properties

Label 2-220-5.2-c2-0-9
Degree 22
Conductor 220220
Sign −0.917+0.398i-0.917 + 0.398i
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  + (−0.176 + 0.176i)3-s + (0.718 − 4.94i)5-s + (−8.33 − 8.33i)7-s + 8.93i·9-s − 3.31·11-s + (−17.8 + 17.8i)13-s + (0.746 + 0.999i)15-s + (−14.0 − 14.0i)17-s − 14.2i·19-s + 2.94·21-s + (17.1 − 17.1i)23-s + (−23.9 − 7.10i)25-s + (−3.16 − 3.16i)27-s − 9.52i·29-s + 17.4·31-s + ⋯
L(s)  = 1  + (−0.0588 + 0.0588i)3-s + (0.143 − 0.989i)5-s + (−1.19 − 1.19i)7-s + 0.993i·9-s − 0.301·11-s + (−1.37 + 1.37i)13-s + (0.0497 + 0.0666i)15-s + (−0.827 − 0.827i)17-s − 0.749i·19-s + 0.140·21-s + (0.745 − 0.745i)23-s + (−0.958 − 0.284i)25-s + (−0.117 − 0.117i)27-s − 0.328i·29-s + 0.561·31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.917+0.398i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.917+0.398i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.917+0.398i-0.917 + 0.398i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(177,⋅)\chi_{220} (177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.917+0.398i)(2,\ 220,\ (\ :1),\ -0.917 + 0.398i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.111721−0.538109i0.111721 - 0.538109i
L(12)L(\frac12) ≈\approx 0.111721−0.538109i0.111721 - 0.538109i
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+(−0.718+4.94i)T 1 + (-0.718 + 4.94i)T
11 1+3.31T 1 + 3.31T
good3 1+(0.176−0.176i)T−9iT2 1 + (0.176 - 0.176i)T - 9iT^{2}
7 1+(8.33+8.33i)T+49iT2 1 + (8.33 + 8.33i)T + 49iT^{2}
13 1+(17.8−17.8i)T−169iT2 1 + (17.8 - 17.8i)T - 169iT^{2}
17 1+(14.0+14.0i)T+289iT2 1 + (14.0 + 14.0i)T + 289iT^{2}
19 1+14.2iT−361T2 1 + 14.2iT - 361T^{2}
23 1+(−17.1+17.1i)T−529iT2 1 + (-17.1 + 17.1i)T - 529iT^{2}
29 1+9.52iT−841T2 1 + 9.52iT - 841T^{2}
31 1−17.4T+961T2 1 - 17.4T + 961T^{2}
37 1+(15.8+15.8i)T+1.36e3iT2 1 + (15.8 + 15.8i)T + 1.36e3iT^{2}
41 1+40.5T+1.68e3T2 1 + 40.5T + 1.68e3T^{2}
43 1+(−26.3+26.3i)T−1.84e3iT2 1 + (-26.3 + 26.3i)T - 1.84e3iT^{2}
47 1+(−19.5−19.5i)T+2.20e3iT2 1 + (-19.5 - 19.5i)T + 2.20e3iT^{2}
53 1+(−25.7+25.7i)T−2.80e3iT2 1 + (-25.7 + 25.7i)T - 2.80e3iT^{2}
59 1−24.2iT−3.48e3T2 1 - 24.2iT - 3.48e3T^{2}
61 1−77.6T+3.72e3T2 1 - 77.6T + 3.72e3T^{2}
67 1+(68.1+68.1i)T+4.48e3iT2 1 + (68.1 + 68.1i)T + 4.48e3iT^{2}
71 1−28.0T+5.04e3T2 1 - 28.0T + 5.04e3T^{2}
73 1+(1.22−1.22i)T−5.32e3iT2 1 + (1.22 - 1.22i)T - 5.32e3iT^{2}
79 1+127.iT−6.24e3T2 1 + 127. iT - 6.24e3T^{2}
83 1+(64.3−64.3i)T−6.88e3iT2 1 + (64.3 - 64.3i)T - 6.88e3iT^{2}
89 1−125.iT−7.92e3T2 1 - 125. iT - 7.92e3T^{2}
97 1+(25.6+25.6i)T+9.40e3iT2 1 + (25.6 + 25.6i)T + 9.40e3iT^{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

−11.69549631069812987668484696235, −10.53343723587865378408380036985, −9.667978947830502891821468892125, −8.875909963959189587043908866140, −7.40822400235407898023688984915, −6.72496719678979424413414156595, −5.00052219434141497326264898737, −4.32181250873677052903059426616, −2.41352739182780778672229813905, −0.28106233884226743977471888162, 2.59077708475202347540553384202, 3.42420094505077448539743587870, 5.49555824800514863130660829985, 6.30722495969737343789055340827, 7.25902842967036021110491370762, 8.644025451142753630943138114588, 9.761207458260218949542535672457, 10.29438564479458356619804454834, 11.66493028900231651552554760636, 12.52574111516389284231544908018

Graph of the ZZ-function along the critical line