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Properties

Label 2-220-5.3-c2-0-7
Degree $2$
Conductor $220$
Sign $-0.0983 + 0.995i$
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  + (−1.52 − 1.52i)3-s + (4.49 − 2.19i)5-s + (0.0256 − 0.0256i)7-s − 4.34i·9-s − 3.31·11-s + (1.36 + 1.36i)13-s + (−10.1 − 3.49i)15-s + (4.75 − 4.75i)17-s − 24.5i·19-s − 0.0783·21-s + (−14.4 − 14.4i)23-s + (15.3 − 19.7i)25-s + (−20.3 + 20.3i)27-s − 51.3i·29-s − 8.20·31-s + ⋯
L(s)  = 1  + (−0.508 − 0.508i)3-s + (0.898 − 0.439i)5-s + (0.00367 − 0.00367i)7-s − 0.483i·9-s − 0.301·11-s + (0.105 + 0.105i)13-s + (−0.679 − 0.233i)15-s + (0.279 − 0.279i)17-s − 1.29i·19-s − 0.00373·21-s + (−0.627 − 0.627i)23-s + (0.613 − 0.789i)25-s + (−0.753 + 0.753i)27-s − 1.76i·29-s − 0.264·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0983 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0983 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.0983 + 0.995i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.0983 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.907864 - 1.00197i\)
\(L(\frac12)\) \(\approx\) \(0.907864 - 1.00197i\)
\(L(2)\) 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 \( 1 + (-4.49 + 2.19i)T \)
11 \( 1 + 3.31T \)
good3 \( 1 + (1.52 + 1.52i)T + 9iT^{2} \)
7 \( 1 + (-0.0256 + 0.0256i)T - 49iT^{2} \)
13 \( 1 + (-1.36 - 1.36i)T + 169iT^{2} \)
17 \( 1 + (-4.75 + 4.75i)T - 289iT^{2} \)
19 \( 1 + 24.5iT - 361T^{2} \)
23 \( 1 + (14.4 + 14.4i)T + 529iT^{2} \)
29 \( 1 + 51.3iT - 841T^{2} \)
31 \( 1 + 8.20T + 961T^{2} \)
37 \( 1 + (0.158 - 0.158i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.4T + 1.68e3T^{2} \)
43 \( 1 + (-38.0 - 38.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (32.5 - 32.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (-57.7 - 57.7i)T + 2.80e3iT^{2} \)
59 \( 1 - 2.32iT - 3.48e3T^{2} \)
61 \( 1 - 34.8T + 3.72e3T^{2} \)
67 \( 1 + (29.3 - 29.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 65.3T + 5.04e3T^{2} \)
73 \( 1 + (29.3 + 29.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 50.5iT - 6.24e3T^{2} \)
83 \( 1 + (-72.6 - 72.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 30.3iT - 7.92e3T^{2} \)
97 \( 1 + (34.2 - 34.2i)T - 9.40e3iT^{2} \)
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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

−11.91685270683465779001222471367, −10.93526355425301732399364033551, −9.748555472394354555200302024111, −9.013958846115846485426793405866, −7.68590943232058061648151946868, −6.45961205693873881823783514113, −5.75045294744345198633370488622, −4.45904295306416114250743502332, −2.50084754765512991761082270705, −0.812285033726378417554554389851, 1.94469228099144606545483090857, 3.63001622212411039576933039594, 5.23360077615689940995688962493, 5.85470060466772574468465048638, 7.18533298848092060924682240898, 8.399447383744524404558912467954, 9.737669495324787618614707479245, 10.37377392293097624841529610842, 11.08114696165531916234712100634, 12.28500413783974897185359889410

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