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Properties

Label 2-220-220.219-c1-0-7
Degree 22
Conductor 220220
Sign 0.829−0.559i0.829 - 0.559i
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  + (−1.24 − 0.664i)2-s + (1.11 + 1.65i)4-s + 2.23·5-s + 4.29i·7-s + (−0.294 − 2.81i)8-s − 3·9-s + (−2.79 − 1.48i)10-s + 3.31i·11-s − 1.90·13-s + (2.85 − 5.36i)14-s + (−1.49 + 3.70i)16-s + 8.08·17-s + (3.74 + 1.99i)18-s + (2.50 + 3.70i)20-s + (2.20 − 4.14i)22-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + 0.999·5-s + 1.62i·7-s + (−0.104 − 0.994i)8-s − 9-s + (−0.882 − 0.469i)10-s + 1.00i·11-s − 0.529·13-s + (0.762 − 1.43i)14-s + (−0.374 + 0.927i)16-s + 1.95·17-s + (0.882 + 0.469i)18-s + (0.559 + 0.829i)20-s + (0.469 − 0.882i)22-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.829−0.559i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.829−0.559i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.829−0.559i0.829 - 0.559i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(219,⋅)\chi_{220} (219, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.829−0.559i)(2,\ 220,\ (\ :1/2),\ 0.829 - 0.559i)

Particular Values

L(1)L(1) ≈\approx 0.851941+0.260365i0.851941 + 0.260365i
L(12)L(\frac12) ≈\approx 0.851941+0.260365i0.851941 + 0.260365i
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+(1.24+0.664i)T 1 + (1.24 + 0.664i)T
5 1−2.23T 1 - 2.23T
11 1−3.31iT 1 - 3.31iT
good3 1+3T2 1 + 3T^{2}
7 1−4.29iT−7T2 1 - 4.29iT - 7T^{2}
13 1+1.90T+13T2 1 + 1.90T + 13T^{2}
17 1−8.08T+17T2 1 - 8.08T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+6.63iT−31T2 1 + 6.63iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1+1.01iT−43T2 1 + 1.01iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1+14.8iT−59T2 1 + 14.8iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1−14.8iT−71T2 1 - 14.8iT - 71T^{2}
73 1+11.8T+73T2 1 + 11.8T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+18.2iT−83T2 1 + 18.2iT - 83T^{2}
89 1−13.4T+89T2 1 - 13.4T + 89T^{2}
97 1−97T2 1 - 97T^{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

−12.19430123383967887211802677783, −11.51799079167151327950356157431, −10.10044111731562785260228317422, −9.538631394086754896917215038881, −8.701017760776357036292952882274, −7.65089402091915618464691992698, −6.16756336670780461942788990235, −5.29526283324559675778006604176, −2.96207303717947277189319206807, −2.02881335765112939592100144208, 1.06519132048910787268098613289, 3.10735027299641714476863282106, 5.23479218386229192692071262718, 6.12772319146569543933992595316, 7.26118771578290698805839930655, 8.193588181776631068815992469664, 9.294705740015931014339870369382, 10.28908412523735357603202407418, 10.73267943137480704734415421868, 11.96858282229378580929994036018

Graph of the ZZ-function along the critical line