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Properties

Label 2-220-4.3-c2-0-35
Degree $2$
Conductor $220$
Sign $-0.170 + 0.985i$
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  + (1.99 + 0.170i)2-s − 4.77i·3-s + (3.94 + 0.680i)4-s − 2.23·5-s + (0.814 − 9.51i)6-s − 5.89i·7-s + (7.73 + 2.02i)8-s − 13.8·9-s + (−4.45 − 0.381i)10-s − 3.31i·11-s + (3.24 − 18.8i)12-s − 6.88·13-s + (1.00 − 11.7i)14-s + 10.6i·15-s + (15.0 + 5.36i)16-s + 14.4·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0853i)2-s − 1.59i·3-s + (0.985 + 0.170i)4-s − 0.447·5-s + (0.135 − 1.58i)6-s − 0.841i·7-s + (0.967 + 0.253i)8-s − 1.53·9-s + (−0.445 − 0.0381i)10-s − 0.301i·11-s + (0.270 − 1.56i)12-s − 0.529·13-s + (0.0717 − 0.838i)14-s + 0.711i·15-s + (0.942 + 0.335i)16-s + 0.852·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.170 + 0.985i$
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: \((2,\ 220,\ (\ :1),\ -0.170 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.67285 - 1.98617i\)
\(L(\frac12)\) \(\approx\) \(1.67285 - 1.98617i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.170i)T \)
5 \( 1 + 2.23T \)
11 \( 1 + 3.31iT \)
good3 \( 1 + 4.77iT - 9T^{2} \)
7 \( 1 + 5.89iT - 49T^{2} \)
13 \( 1 + 6.88T + 169T^{2} \)
17 \( 1 - 14.4T + 289T^{2} \)
19 \( 1 - 19.0iT - 361T^{2} \)
23 \( 1 + 11.4iT - 529T^{2} \)
29 \( 1 - 11.2T + 841T^{2} \)
31 \( 1 + 19.2iT - 961T^{2} \)
37 \( 1 - 68.0T + 1.36e3T^{2} \)
41 \( 1 + 61.6T + 1.68e3T^{2} \)
43 \( 1 - 33.0iT - 1.84e3T^{2} \)
47 \( 1 - 71.8iT - 2.20e3T^{2} \)
53 \( 1 - 81.0T + 2.80e3T^{2} \)
59 \( 1 - 71.0iT - 3.48e3T^{2} \)
61 \( 1 - 23.1T + 3.72e3T^{2} \)
67 \( 1 + 92.8iT - 4.48e3T^{2} \)
71 \( 1 - 21.1iT - 5.04e3T^{2} \)
73 \( 1 + 51.6T + 5.32e3T^{2} \)
79 \( 1 + 22.8iT - 6.24e3T^{2} \)
83 \( 1 - 128. iT - 6.88e3T^{2} \)
89 \( 1 + 142.T + 7.92e3T^{2} \)
97 \( 1 + 102.T + 9.40e3T^{2} \)
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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

−12.11599206848936215503259320688, −11.32296331300684424407139824000, −10.15465067755153266650943847972, −8.101449962802606520342700342351, −7.59401680695461780911681713918, −6.69764099678399828658249576295, −5.73898971462914921152193067512, −4.18169339585684536539096005930, −2.76443687666979398088628738837, −1.15965255500993775673706178197, 2.70626814671429206186014773402, 3.80869979993598703128435276411, 4.88436833251235808867299363686, 5.55286166096533222798143960067, 7.08224560779651400197734030479, 8.526315383270118278687591934842, 9.691838430193790863290891328477, 10.44563983484037099535333760303, 11.54266101904305066529794504974, 12.06034027768591286447195142099

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