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Properties

Label 2-220-44.43-c1-0-10
Degree $2$
Conductor $220$
Sign $0.343 + 0.939i$
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  + (−0.960 − 1.03i)2-s − 0.450i·3-s + (−0.154 + 1.99i)4-s + 5-s + (−0.467 + 0.432i)6-s − 1.25·7-s + (2.21 − 1.75i)8-s + 2.79·9-s + (−0.960 − 1.03i)10-s + (3.01 − 1.37i)11-s + (0.898 + 0.0698i)12-s − 3.65i·13-s + (1.20 + 1.30i)14-s − 0.450i·15-s + (−3.95 − 0.618i)16-s + 1.35i·17-s + ⋯
L(s)  = 1  + (−0.679 − 0.733i)2-s − 0.260i·3-s + (−0.0774 + 0.996i)4-s + 0.447·5-s + (−0.190 + 0.176i)6-s − 0.475·7-s + (0.784 − 0.620i)8-s + 0.932·9-s + (−0.303 − 0.328i)10-s + (0.909 − 0.414i)11-s + (0.259 + 0.0201i)12-s − 1.01i·13-s + (0.322 + 0.349i)14-s − 0.116i·15-s + (−0.987 − 0.154i)16-s + 0.329i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796331 - 0.556926i\)
\(L(\frac12)\) \(\approx\) \(0.796331 - 0.556926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.960 + 1.03i)T \)
5 \( 1 - T \)
11 \( 1 + (-3.01 + 1.37i)T \)
good3 \( 1 + 0.450iT - 3T^{2} \)
7 \( 1 + 1.25T + 7T^{2} \)
13 \( 1 + 3.65iT - 13T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 + 3.06iT - 23T^{2} \)
29 \( 1 - 2.29iT - 29T^{2} \)
31 \( 1 + 5.36iT - 31T^{2} \)
37 \( 1 + 3.17T + 37T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 - 4.16T + 43T^{2} \)
47 \( 1 - 9.18iT - 47T^{2} \)
53 \( 1 - 4.41T + 53T^{2} \)
59 \( 1 - 8.87iT - 59T^{2} \)
61 \( 1 - 5.01iT - 61T^{2} \)
67 \( 1 - 9.46iT - 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 6.68T + 83T^{2} \)
89 \( 1 + 9.39T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−12.14040112385413646575903457759, −10.99263820114980407191616391263, −10.08925262041510538407858183099, −9.386297445177562480619476470007, −8.287971834615081924493340883407, −7.20304791236222371755833824330, −6.11440847365051470276656342851, −4.27632374393016544779193315144, −2.94324033466690018803420983697, −1.25823738211269500436308085280, 1.69148623158413078306913117523, 4.06603420002457643688347955046, 5.33798880440075497666870362337, 6.69731861199770789060533113746, 7.16427773524480017182779222821, 8.750811927904643496487863185441, 9.552868509545512698419348275934, 10.07676086243640397745089178157, 11.30641115083137010437546634956, 12.47840139716684268869231767815

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