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Properties

Label 2-220-11.7-c2-0-4
Degree $2$
Conductor $220$
Sign $0.555 + 0.831i$
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  + (0.486 − 1.49i)3-s + (1.80 − 1.31i)5-s + (−2.66 + 0.866i)7-s + (5.27 + 3.83i)9-s + (8.69 − 6.73i)11-s + (11.7 − 16.1i)13-s + (−1.08 − 3.34i)15-s + (−6.11 − 8.41i)17-s + (−14.5 − 4.71i)19-s + 4.41i·21-s − 9.47·23-s + (1.54 − 4.75i)25-s + (19.7 − 14.3i)27-s + (29.9 − 9.72i)29-s + (8.91 + 6.48i)31-s + ⋯
L(s)  = 1  + (0.162 − 0.498i)3-s + (0.361 − 0.262i)5-s + (−0.380 + 0.123i)7-s + (0.586 + 0.425i)9-s + (0.790 − 0.612i)11-s + (0.903 − 1.24i)13-s + (−0.0725 − 0.223i)15-s + (−0.359 − 0.495i)17-s + (−0.763 − 0.248i)19-s + 0.210i·21-s − 0.412·23-s + (0.0618 − 0.190i)25-s + (0.732 − 0.531i)27-s + (1.03 − 0.335i)29-s + (0.287 + 0.209i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.555 + 0.831i$
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: \((2,\ 220,\ (\ :1),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56769 - 0.837831i\)
\(L(\frac12)\) \(\approx\) \(1.56769 - 0.837831i\)
\(L(2)\) 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 \( 1 + (-8.69 + 6.73i)T \)
good3 \( 1 + (-0.486 + 1.49i)T + (-7.28 - 5.29i)T^{2} \)
7 \( 1 + (2.66 - 0.866i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (-11.7 + 16.1i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (6.11 + 8.41i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (14.5 + 4.71i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 9.47T + 529T^{2} \)
29 \( 1 + (-29.9 + 9.72i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-8.91 - 6.48i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-5.25 - 16.1i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-20.2 - 6.57i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 31.6iT - 1.84e3T^{2} \)
47 \( 1 + (24.8 - 76.4i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (20.0 + 14.5i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-17.3 - 53.5i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-63.7 - 87.7i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 46.4T + 4.48e3T^{2} \)
71 \( 1 + (100. - 73.1i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-68.5 + 22.2i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (13.8 - 19.0i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (53.1 + 73.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 32.7T + 7.92e3T^{2} \)
97 \( 1 + (-141. - 102. i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−12.06533665902709290337209984801, −10.89137848299368771925774062810, −9.993160071179339568340435601913, −8.812600125998483508391382891019, −8.026265284960473161705390675380, −6.70277240320703914639663474206, −5.86628627248063063567155138939, −4.37376555881255639784126871618, −2.81342781870855316655188698518, −1.11981098770261820291371885842, 1.75962102442548852314490686115, 3.67415992569808480698762375598, 4.48242396732470774410814437199, 6.34167527255720959813925480615, 6.80548628651854089371832477898, 8.497201937083234351672761494717, 9.414766839561091579816116133069, 10.10962297353996702608535465411, 11.14610047969378062006552629711, 12.23074238837117137918573120281

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