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Properties

Label 2-220-55.24-c2-0-8
Degree $2$
Conductor $220$
Sign $0.0194 + 0.999i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 1.83i)3-s + (−3.12 − 3.90i)5-s + (−5.02 − 3.65i)7-s + (1.19 − 3.68i)9-s + (−6.42 − 8.92i)11-s + (5.44 − 16.7i)13-s + (2.99 − 10.9i)15-s + (9.49 + 29.2i)17-s + (−11.3 − 15.5i)19-s − 14.0i·21-s − 8.32i·23-s + (−5.50 + 24.3i)25-s + (27.7 − 9.00i)27-s + (22.9 − 31.5i)29-s + (−7.39 + 22.7i)31-s + ⋯
L(s)  = 1  + (0.443 + 0.610i)3-s + (−0.624 − 0.781i)5-s + (−0.718 − 0.522i)7-s + (0.133 − 0.409i)9-s + (−0.584 − 0.811i)11-s + (0.418 − 1.28i)13-s + (0.199 − 0.727i)15-s + (0.558 + 1.71i)17-s + (−0.596 − 0.820i)19-s − 0.670i·21-s − 0.362i·23-s + (−0.220 + 0.975i)25-s + (1.02 − 0.333i)27-s + (0.790 − 1.08i)29-s + (−0.238 + 0.734i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0194 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0194 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.0194 + 0.999i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ 0.0194 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.835687 - 0.819601i\)
\(L(\frac12)\) \(\approx\) \(0.835687 - 0.819601i\)
\(L(2)\) 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 \( 1 + (3.12 + 3.90i)T \)
11 \( 1 + (6.42 + 8.92i)T \)
good3 \( 1 + (-1.33 - 1.83i)T + (-2.78 + 8.55i)T^{2} \)
7 \( 1 + (5.02 + 3.65i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-5.44 + 16.7i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-9.49 - 29.2i)T + (-233. + 169. i)T^{2} \)
19 \( 1 + (11.3 + 15.5i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 8.32iT - 529T^{2} \)
29 \( 1 + (-22.9 + 31.5i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (7.39 - 22.7i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (12.3 - 16.9i)T + (-423. - 1.30e3i)T^{2} \)
41 \( 1 + (15.5 + 21.3i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 58.0T + 1.84e3T^{2} \)
47 \( 1 + (53.0 + 73.0i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-76.2 - 24.7i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-55.1 - 40.0i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-41.2 + 13.4i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 78.6iT - 4.48e3T^{2} \)
71 \( 1 + (-21.9 - 67.5i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (47.8 + 34.8i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-51.1 - 16.6i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-20.7 - 63.9i)T + (-5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 166.T + 7.92e3T^{2} \)
97 \( 1 + (84.5 + 27.4i)T + (7.61e3 + 5.53e3i)T^{2} \)
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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

−11.91681301396373693637807732967, −10.51144232125251125525327569764, −10.07774695288841803989105694644, −8.485458243984675225323240730349, −8.373649991438213373560945797340, −6.69684836400045615340093746373, −5.41280979769854528323007372564, −3.98719814228007108986439935678, −3.26563238293686016543238975095, −0.60898337750792502386558023175, 2.12838414275815792247537063762, 3.28472824161627183558972096233, 4.84837149044965469477774290752, 6.51873241364309885139966683023, 7.23721067884943552844727501560, 8.130890577067849817169774739748, 9.366826357469607181609904061472, 10.31979918382304950742501116973, 11.51257722940730551409064872488, 12.27882437479974243972676828537

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