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Properties

Label 2-220-11.8-c2-0-3
Degree $2$
Conductor $220$
Sign $-0.282 - 0.959i$
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  + (1.75 + 5.41i)3-s + (1.80 + 1.31i)5-s + (10.9 + 3.57i)7-s + (−18.9 + 13.7i)9-s + (10.7 − 2.36i)11-s + (−10.9 − 15.0i)13-s + (−3.93 + 12.1i)15-s + (10.4 − 14.3i)17-s + (2.88 − 0.938i)19-s + 65.8i·21-s − 27.6·23-s + (1.54 + 4.75i)25-s + (−66.3 − 48.2i)27-s + (−25.0 − 8.13i)29-s + (−1.18 + 0.857i)31-s + ⋯
L(s)  = 1  + (0.586 + 1.80i)3-s + (0.361 + 0.262i)5-s + (1.57 + 0.510i)7-s + (−2.10 + 1.52i)9-s + (0.976 − 0.215i)11-s + (−0.839 − 1.15i)13-s + (−0.262 + 0.807i)15-s + (0.614 − 0.846i)17-s + (0.151 − 0.0493i)19-s + 3.13i·21-s − 1.20·23-s + (0.0618 + 0.190i)25-s + (−2.45 − 1.78i)27-s + (−0.863 − 0.280i)29-s + (−0.0380 + 0.0276i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.282 - 0.959i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.282 - 0.959i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30516 + 1.74542i\)
\(L(\frac12)\) \(\approx\) \(1.30516 + 1.74542i\)
\(L(2)\) 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 \( 1 + (-10.7 + 2.36i)T \)
good3 \( 1 + (-1.75 - 5.41i)T + (-7.28 + 5.29i)T^{2} \)
7 \( 1 + (-10.9 - 3.57i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (10.9 + 15.0i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-10.4 + 14.3i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-2.88 + 0.938i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 27.6T + 529T^{2} \)
29 \( 1 + (25.0 + 8.13i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (1.18 - 0.857i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (6.74 - 20.7i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-26.1 + 8.48i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 13.8iT - 1.84e3T^{2} \)
47 \( 1 + (19.8 + 61.0i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-48.9 + 35.5i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (5.59 - 17.2i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-2.16 + 2.98i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 120.T + 4.48e3T^{2} \)
71 \( 1 + (19.1 + 13.8i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (21.9 + 7.13i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (9.48 + 13.0i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (9.03 - 12.4i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 29.0T + 7.92e3T^{2} \)
97 \( 1 + (113. - 82.4i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.91691673580072363511967276769, −11.24881984353581194992788143082, −10.22048043462354510355794854456, −9.556792175686729542022468124666, −8.563184011217841390589617447798, −7.74721058537836266068992037151, −5.58905855510901117051426471314, −4.96318742100982413404894768749, −3.73295177269678469215917468592, −2.39577305616518531718662898701, 1.37847923103373789835956288199, 2.03612145899403705179530304445, 4.13111175967788032788332826219, 5.77209160404461273127212811754, 6.94810664088593379370808753571, 7.70127278312189792925924805643, 8.502581798638012575228792303386, 9.525803819509458684248874057042, 11.21646688905852255703144768162, 11.99492196721642102086921761907

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