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Properties

Label 2-220-55.54-c0-0-1
Degree $2$
Conductor $220$
Sign $0.5 + 0.866i$
Analytic cond. $0.109794$
Root an. cond. $0.331352$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + (0.5 + 0.866i)5-s − 1.99·9-s − 11-s + (1.49 − 0.866i)15-s + 1.73i·23-s + (−0.499 + 0.866i)25-s + 1.73i·27-s + 31-s + 1.73i·33-s − 1.73i·37-s + (−0.999 − 1.73i)45-s − 49-s + (−0.5 − 0.866i)55-s − 59-s + ⋯
L(s)  = 1  − 1.73i·3-s + (0.5 + 0.866i)5-s − 1.99·9-s − 11-s + (1.49 − 0.866i)15-s + 1.73i·23-s + (−0.499 + 0.866i)25-s + 1.73i·27-s + 31-s + 1.73i·33-s − 1.73i·37-s + (−0.999 − 1.73i)45-s − 49-s + (−0.5 − 0.866i)55-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(0.109794\)
Root analytic conductor: \(0.331352\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :0),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7222949574\)
\(L(\frac12)\) \(\approx\) \(0.7222949574\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
good3 \( 1 + 1.73iT - T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−12.50076087189891172453642928629, −11.52341064033869578303731870361, −10.64624087594402505418448670014, −9.378351762009847898448380875780, −7.940193807886545502325381459736, −7.37578574256180562023732265706, −6.36422800796378950164003148517, −5.48699164564002589839083031839, −3.06636432204313608798296373661, −1.90512182955237445659925190525, 2.83883745991311174958759578065, 4.44806780003700737466445685372, 5.01816571985453217309847345593, 6.16819194162180967358625418538, 8.195590797707479658159659135353, 8.880042955447999587174155591469, 10.00039946931955901204771160351, 10.35125654164405097163988718078, 11.53086444641975556756976949227, 12.68141026790170625600555325741

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