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Properties

Label 2-220-55.2-c1-0-0
Degree $2$
Conductor $220$
Sign $-0.632 - 0.774i$
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  + (−0.467 + 2.95i)3-s + (−0.111 + 2.23i)5-s + (2.35 − 0.372i)7-s + (−5.63 − 1.83i)9-s + (0.368 − 3.29i)11-s + (−4.91 + 2.50i)13-s + (−6.53 − 1.37i)15-s + (5.13 + 2.61i)17-s + (−2.00 + 1.45i)19-s + 7.11i·21-s + (4.85 − 4.85i)23-s + (−4.97 − 0.499i)25-s + (3.96 − 7.78i)27-s + (4.14 + 3.01i)29-s + (0.457 − 1.40i)31-s + ⋯
L(s)  = 1  + (−0.269 + 1.70i)3-s + (−0.0500 + 0.998i)5-s + (0.889 − 0.140i)7-s + (−1.87 − 0.610i)9-s + (0.111 − 0.993i)11-s + (−1.36 + 0.694i)13-s + (−1.68 − 0.354i)15-s + (1.24 + 0.634i)17-s + (−0.460 + 0.334i)19-s + 1.55i·21-s + (1.01 − 1.01i)23-s + (−0.994 − 0.0999i)25-s + (0.762 − 1.49i)27-s + (0.770 + 0.559i)29-s + (0.0821 − 0.252i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472664 + 0.995593i\)
\(L(\frac12)\) \(\approx\) \(0.472664 + 0.995593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.111 - 2.23i)T \)
11 \( 1 + (-0.368 + 3.29i)T \)
good3 \( 1 + (0.467 - 2.95i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (-2.35 + 0.372i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (4.91 - 2.50i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-5.13 - 2.61i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (2.00 - 1.45i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.85 + 4.85i)T - 23iT^{2} \)
29 \( 1 + (-4.14 - 3.01i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.457 + 1.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.103 + 0.654i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-0.176 - 0.242i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.73 - 6.73i)T + 43iT^{2} \)
47 \( 1 + (-7.46 - 1.18i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.877 + 1.72i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-0.419 + 0.576i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.32 + 1.73i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.03 + 6.03i)T + 67iT^{2} \)
71 \( 1 + (1.48 + 4.56i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.311 - 1.96i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.20 + 6.79i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.000475 - 0.000934i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 4.41iT - 89T^{2} \)
97 \( 1 + (7.94 - 4.05i)T + (57.0 - 78.4i)T^{2} \)
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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

−12.25975165966916893169898226796, −11.27004657125575571835257821617, −10.68494472462660724429169994890, −9.993195851111424057436043227681, −8.921390793847294828500506837242, −7.76431224996133310476189314991, −6.28274525462595191127994662185, −5.10840908601421569395961569610, −4.11992434092538025913907161252, −2.91066225364259885643074043508, 1.04498458614905352770579623825, 2.37203406668290061412555987852, 4.85674927802972792129147083606, 5.60163590168205481018852504757, 7.26316351367758729295661453062, 7.61954850139093181639833430232, 8.686462829713609385039696736573, 9.929307734426621441948663210516, 11.48790785070983068499519380221, 12.23415660596411427188098462316

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