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Properties

Label 2-220-4.3-c2-0-39
Degree $2$
Conductor $220$
Sign $-0.00238 - 0.999i$
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  + (−0.00238 − 1.99i)2-s − 5.13i·3-s + (−3.99 + 0.00955i)4-s − 2.23·5-s + (−10.2 + 0.0122i)6-s + 6.37i·7-s + (0.0286 + 7.99i)8-s − 17.3·9-s + (0.00533 + 4.47i)10-s − 3.31i·11-s + (0.0490 + 20.5i)12-s − 1.30·13-s + (12.7 − 0.0152i)14-s + 11.4i·15-s + (15.9 − 0.0764i)16-s − 24.9·17-s + ⋯
L(s)  = 1  + (−0.00119 − 0.999i)2-s − 1.71i·3-s + (−0.999 + 0.00238i)4-s − 0.447·5-s + (−1.71 + 0.00204i)6-s + 0.911i·7-s + (0.00358 + 0.999i)8-s − 1.92·9-s + (0.000533 + 0.447i)10-s − 0.301i·11-s + (0.00408 + 1.71i)12-s − 0.100·13-s + (0.911 − 0.00108i)14-s + 0.765i·15-s + (0.999 − 0.00477i)16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00238 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.00238 - 0.999i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.00238 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.359678 + 0.360538i\)
\(L(\frac12)\) \(\approx\) \(0.359678 + 0.360538i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.00238 + 1.99i)T \)
5 \( 1 + 2.23T \)
11 \( 1 + 3.31iT \)
good3 \( 1 + 5.13iT - 9T^{2} \)
7 \( 1 - 6.37iT - 49T^{2} \)
13 \( 1 + 1.30T + 169T^{2} \)
17 \( 1 + 24.9T + 289T^{2} \)
19 \( 1 + 29.5iT - 361T^{2} \)
23 \( 1 + 4.05iT - 529T^{2} \)
29 \( 1 + 6.14T + 841T^{2} \)
31 \( 1 - 16.6iT - 961T^{2} \)
37 \( 1 + 11.3T + 1.36e3T^{2} \)
41 \( 1 + 46.1T + 1.68e3T^{2} \)
43 \( 1 + 51.0iT - 1.84e3T^{2} \)
47 \( 1 - 74.9iT - 2.20e3T^{2} \)
53 \( 1 + 52.6T + 2.80e3T^{2} \)
59 \( 1 + 11.6iT - 3.48e3T^{2} \)
61 \( 1 - 62.5T + 3.72e3T^{2} \)
67 \( 1 - 49.7iT - 4.48e3T^{2} \)
71 \( 1 + 132. iT - 5.04e3T^{2} \)
73 \( 1 + 113.T + 5.32e3T^{2} \)
79 \( 1 + 144. iT - 6.24e3T^{2} \)
83 \( 1 + 67.6iT - 6.88e3T^{2} \)
89 \( 1 - 155.T + 7.92e3T^{2} \)
97 \( 1 - 163.T + 9.40e3T^{2} \)
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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

−11.61815384051976481170169391782, −10.83662427128112166005017928592, −9.007472313709219695901987548787, −8.581109872283463375857655090658, −7.34468208222787535723996621280, −6.23859186772818197063499473546, −4.86548648556205105436093575754, −2.94902107882452218126441279127, −1.95692677359588519435533378752, −0.27484814452698873027877878486, 3.70173936888732294773416110379, 4.29124969610334717819083466318, 5.30424949253546613414363734875, 6.66381905477249484998235442595, 7.917009508465619784516412199275, 8.860662321637006978316495303093, 9.873790091767117191972210169076, 10.46678357748583220625832211322, 11.58002733848919251756750290510, 13.04776186922666598459539412531

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