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Properties

Label 2-220-44.43-c1-0-21
Degree $2$
Conductor $220$
Sign $-0.859 - 0.511i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 0.5i)2-s − 3.25i·3-s + (1.50 + 1.32i)4-s − 5-s + (−1.62 + 4.31i)6-s − 3.25·7-s + (−1.32 − 2.50i)8-s − 7.62·9-s + (1.32 + 0.5i)10-s + (3.25 − 0.613i)11-s + (4.31 − 4.88i)12-s + 2i·13-s + (4.31 + 1.62i)14-s + 3.25i·15-s + (0.500 + 3.96i)16-s − 4.62i·17-s + ⋯
L(s)  = 1  + (−0.935 − 0.353i)2-s − 1.88i·3-s + (0.750 + 0.661i)4-s − 0.447·5-s + (−0.665 + 1.76i)6-s − 1.23·7-s + (−0.467 − 0.883i)8-s − 2.54·9-s + (0.418 + 0.158i)10-s + (0.982 − 0.185i)11-s + (1.24 − 1.41i)12-s + 0.554i·13-s + (1.15 + 0.435i)14-s + 0.841i·15-s + (0.125 + 0.992i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106636 + 0.387833i\)
\(L(\frac12)\) \(\approx\) \(0.106636 + 0.387833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.5i)T \)
5 \( 1 + T \)
11 \( 1 + (-3.25 + 0.613i)T \)
good3 \( 1 + 3.25iT - 3T^{2} \)
7 \( 1 + 3.25T + 7T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 4.62iT - 17T^{2} \)
19 \( 1 + 4.48T + 19T^{2} \)
23 \( 1 - 1.22iT - 23T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + 3.25iT - 31T^{2} \)
37 \( 1 + 4.62T + 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 - 1.22T + 43T^{2} \)
47 \( 1 - 1.22iT - 47T^{2} \)
53 \( 1 - 0.623T + 53T^{2} \)
59 \( 1 + 11.8iT - 59T^{2} \)
61 \( 1 + 1.37iT - 61T^{2} \)
67 \( 1 + 7.74iT - 67T^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 + 8.62T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−11.89415353977896780908899112575, −11.04714208181117930799686387740, −9.462324678097205844554466460950, −8.717990445969213562244980623087, −7.61355778544448059297534701619, −6.78532184450892879307294524159, −6.26133455960287099417198064282, −3.44574749391459005168799810958, −2.11771274823639781110416119215, −0.43139058921706822383338988331, 3.13452885624797282855874462960, 4.23848750455116035376804384910, 5.73812589338021216771892245360, 6.69049441130634115522561670886, 8.422039851264062879159087937795, 9.009589749303747914626212135006, 10.00001300379123201288986180459, 10.46418871091837657108211842976, 11.40848017235116665776792906963, 12.58144531700598802008619767178

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