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Properties

Label 2-220-44.19-c1-0-16
Degree $2$
Conductor $220$
Sign $0.944 - 0.327i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.478 + 1.33i)2-s + (2.28 − 0.743i)3-s + (−1.54 − 1.27i)4-s + (0.809 + 0.587i)5-s + (−0.104 + 3.39i)6-s + (1.53 − 4.71i)7-s + (2.43 − 1.44i)8-s + (2.25 − 1.63i)9-s + (−1.16 + 0.795i)10-s + (−2.66 + 1.97i)11-s + (−4.47 − 1.76i)12-s + (0.446 + 0.614i)13-s + (5.54 + 4.29i)14-s + (2.28 + 0.743i)15-s + (0.760 + 3.92i)16-s + (−1.55 + 2.14i)17-s + ⋯
L(s)  = 1  + (−0.338 + 0.941i)2-s + (1.32 − 0.429i)3-s + (−0.771 − 0.636i)4-s + (0.361 + 0.262i)5-s + (−0.0426 + 1.38i)6-s + (0.578 − 1.78i)7-s + (0.859 − 0.510i)8-s + (0.750 − 0.545i)9-s + (−0.369 + 0.251i)10-s + (−0.802 + 0.596i)11-s + (−1.29 − 0.509i)12-s + (0.123 + 0.170i)13-s + (1.48 + 1.14i)14-s + (0.590 + 0.191i)15-s + (0.190 + 0.981i)16-s + (−0.377 + 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48778 + 0.250510i\)
\(L(\frac12)\) \(\approx\) \(1.48778 + 0.250510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.478 - 1.33i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (2.66 - 1.97i)T \)
good3 \( 1 + (-2.28 + 0.743i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.53 + 4.71i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.446 - 0.614i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.55 - 2.14i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.730 + 2.24i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.25iT - 23T^{2} \)
29 \( 1 + (2.81 + 0.915i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.03 - 5.54i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.119 + 0.367i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.54 + 0.826i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.07T + 43T^{2} \)
47 \( 1 + (-1.30 + 0.423i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.51 + 3.27i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (12.1 + 3.94i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.02 + 5.53i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.0695iT - 67T^{2} \)
71 \( 1 + (-9.22 + 12.7i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.13 + 1.02i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.948 - 0.688i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.79 + 1.30i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.02T + 89T^{2} \)
97 \( 1 + (11.6 - 8.44i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.06352320053508030265970071685, −10.96911509291583063577226578741, −10.13987388155976024939351603318, −9.204258537047561690363113717632, −8.018864815245938071800547894217, −7.52450336992040622216120912567, −6.71393165598053133342708694002, −4.96130055507239827965535109454, −3.69470324510167727779705716799, −1.68325905157608750826784759236, 2.26720114673310848632125787792, 2.85893416084571325850330162329, 4.48924300063501911734850840226, 5.70556644661589246090367214330, 8.032602044275285093340444419186, 8.551780340112403005092922955333, 9.154940003541033023848503856673, 10.10178748790542630059140134725, 11.22617713773944432457329229821, 12.24392296603767026255277892440

Graph of the $Z$-function along the critical line