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Properties

Label 2-220-11.7-c2-0-1
Degree $2$
Conductor $220$
Sign $-0.971 - 0.236i$
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  + (−1.08 + 3.34i)3-s + (1.80 − 1.31i)5-s + (−6.54 + 2.12i)7-s + (−2.72 − 1.97i)9-s + (−1.85 + 10.8i)11-s + (−6.20 + 8.54i)13-s + (2.42 + 7.47i)15-s + (−14.5 − 19.9i)17-s + (−2.90 − 0.944i)19-s − 24.2i·21-s − 28.7·23-s + (1.54 − 4.75i)25-s + (−16.0 + 11.6i)27-s + (−14.4 + 4.70i)29-s + (22.8 + 16.6i)31-s + ⋯
L(s)  = 1  + (−0.362 + 1.11i)3-s + (0.361 − 0.262i)5-s + (−0.935 + 0.303i)7-s + (−0.302 − 0.219i)9-s + (−0.168 + 0.985i)11-s + (−0.477 + 0.657i)13-s + (0.161 + 0.498i)15-s + (−0.853 − 1.17i)17-s + (−0.153 − 0.0497i)19-s − 1.15i·21-s − 1.25·23-s + (0.0618 − 0.190i)25-s + (−0.593 + 0.431i)27-s + (−0.499 + 0.162i)29-s + (0.738 + 0.536i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.971 - 0.236i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.971 - 0.236i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0899299 + 0.748364i\)
\(L(\frac12)\) \(\approx\) \(0.0899299 + 0.748364i\)
\(L(2)\) 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 \( 1 + (-1.80 + 1.31i)T \)
11 \( 1 + (1.85 - 10.8i)T \)
good3 \( 1 + (1.08 - 3.34i)T + (-7.28 - 5.29i)T^{2} \)
7 \( 1 + (6.54 - 2.12i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (6.20 - 8.54i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (14.5 + 19.9i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (2.90 + 0.944i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 28.7T + 529T^{2} \)
29 \( 1 + (14.4 - 4.70i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-22.8 - 16.6i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (0.847 + 2.60i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-58.6 - 19.0i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 40.5iT - 1.84e3T^{2} \)
47 \( 1 + (-7.00 + 21.5i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-69.9 - 50.8i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-22.7 - 70.1i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (34.2 + 47.1i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 64.6T + 4.48e3T^{2} \)
71 \( 1 + (-49.4 + 35.9i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (80.2 - 26.0i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (44.8 - 61.7i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (11.1 + 15.2i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 127.T + 7.92e3T^{2} \)
97 \( 1 + (-71.6 - 52.0i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−12.42379773502182527248776974467, −11.50039546796559675203740811348, −10.30036224972609564783318741150, −9.639728843558179518731744330736, −9.077763667478264109541978473134, −7.36560450364445617277856966645, −6.22288993831659602584989354865, −4.97930615329424284221507119874, −4.17451400514341963007251869338, −2.44038482446670441936589880759, 0.40749603546298301948403356511, 2.24832094433324393884940285935, 3.80656936747260255670372073061, 5.83968630792547546966490594910, 6.35097026063500771315932683938, 7.41741293294075849502207595229, 8.445462617608429071025704391246, 9.819338775015604086925921171295, 10.65488336558283956703400403032, 11.75972917665926412346251118152

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