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Properties

Label 2-220-20.3-c1-0-25
Degree $2$
Conductor $220$
Sign $-0.620 + 0.783i$
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  + (−1.41 − 0.102i)2-s + (2.05 − 2.05i)3-s + (1.97 + 0.289i)4-s + (−2.23 + 0.0646i)5-s + (−3.10 + 2.68i)6-s + (−3.00 − 3.00i)7-s + (−2.76 − 0.611i)8-s − 5.40i·9-s + (3.15 + 0.138i)10-s + i·11-s + (4.65 − 3.46i)12-s + (−0.709 − 0.709i)13-s + (3.93 + 4.55i)14-s + (−4.45 + 4.71i)15-s + (3.83 + 1.14i)16-s + (−1.02 + 1.02i)17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0725i)2-s + (1.18 − 1.18i)3-s + (0.989 + 0.144i)4-s + (−0.999 + 0.0289i)5-s + (−1.26 + 1.09i)6-s + (−1.13 − 1.13i)7-s + (−0.976 − 0.216i)8-s − 1.80i·9-s + (0.999 + 0.0436i)10-s + 0.301i·11-s + (1.34 − 0.999i)12-s + (−0.196 − 0.196i)13-s + (1.05 + 1.21i)14-s + (−1.14 + 1.21i)15-s + (0.958 + 0.286i)16-s + (−0.249 + 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.620 + 0.783i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.620 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352386 - 0.728608i\)
\(L(\frac12)\) \(\approx\) \(0.352386 - 0.728608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.102i)T \)
5 \( 1 + (2.23 - 0.0646i)T \)
11 \( 1 - iT \)
good3 \( 1 + (-2.05 + 2.05i)T - 3iT^{2} \)
7 \( 1 + (3.00 + 3.00i)T + 7iT^{2} \)
13 \( 1 + (0.709 + 0.709i)T + 13iT^{2} \)
17 \( 1 + (1.02 - 1.02i)T - 17iT^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 + (-3.28 + 3.28i)T - 23iT^{2} \)
29 \( 1 - 0.344iT - 29T^{2} \)
31 \( 1 + 5.97iT - 31T^{2} \)
37 \( 1 + (-1.57 + 1.57i)T - 37iT^{2} \)
41 \( 1 + 6.42T + 41T^{2} \)
43 \( 1 + (-3.02 + 3.02i)T - 43iT^{2} \)
47 \( 1 + (-6.62 - 6.62i)T + 47iT^{2} \)
53 \( 1 + (-1.92 - 1.92i)T + 53iT^{2} \)
59 \( 1 + 1.69T + 59T^{2} \)
61 \( 1 - 2.56T + 61T^{2} \)
67 \( 1 + (7.39 + 7.39i)T + 67iT^{2} \)
71 \( 1 - 4.24iT - 71T^{2} \)
73 \( 1 + (-11.1 - 11.1i)T + 73iT^{2} \)
79 \( 1 - 0.182T + 79T^{2} \)
83 \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \)
89 \( 1 + 6.52iT - 89T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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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

−12.10079403137539487118232985640, −10.85852198181053378217115226538, −9.740853130552430530646884909319, −8.852446690246175177725226805483, −7.73888329944640377411117637950, −7.32107843769141394554869773564, −6.54527904030648922201817437469, −3.72150194684757832624412693329, −2.75711923188480933810815012183, −0.821591325168649710468636173725, 2.81115472296367001496559421709, 3.47296389896765139012614688578, 5.27522983433469860471268931310, 6.92607924721172206211989401623, 8.060400769657386415295319361124, 8.983124638889303783626531154981, 9.360169832523437352903694066470, 10.33037277044914046707075342142, 11.47344530612160800433812426829, 12.32493971681040434983847790571

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