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Properties

Label 2-220-20.3-c1-0-15
Degree $2$
Conductor $220$
Sign $0.764 + 0.644i$
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  + (−1.40 + 0.171i)2-s + (1.44 − 1.44i)3-s + (1.94 − 0.481i)4-s + (0.849 − 2.06i)5-s + (−1.77 + 2.27i)6-s + (3.60 + 3.60i)7-s + (−2.64 + 1.00i)8-s − 1.16i·9-s + (−0.837 + 3.04i)10-s − i·11-s + (2.10 − 3.49i)12-s + (−2.27 − 2.27i)13-s + (−5.67 − 4.44i)14-s + (−1.76 − 4.21i)15-s + (3.53 − 1.86i)16-s + (−3.21 + 3.21i)17-s + ⋯
L(s)  = 1  + (−0.992 + 0.121i)2-s + (0.833 − 0.833i)3-s + (0.970 − 0.240i)4-s + (0.379 − 0.925i)5-s + (−0.726 + 0.928i)6-s + (1.36 + 1.36i)7-s + (−0.934 + 0.356i)8-s − 0.388i·9-s + (−0.264 + 0.964i)10-s − 0.301i·11-s + (0.608 − 1.00i)12-s + (−0.630 − 0.630i)13-s + (−1.51 − 1.18i)14-s + (−0.454 − 1.08i)15-s + (0.884 − 0.467i)16-s + (−0.780 + 0.780i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.764 + 0.644i$
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: \((2,\ 220,\ (\ :1/2),\ 0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11131 - 0.405586i\)
\(L(\frac12)\) \(\approx\) \(1.11131 - 0.405586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.171i)T \)
5 \( 1 + (-0.849 + 2.06i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-1.44 + 1.44i)T - 3iT^{2} \)
7 \( 1 + (-3.60 - 3.60i)T + 7iT^{2} \)
13 \( 1 + (2.27 + 2.27i)T + 13iT^{2} \)
17 \( 1 + (3.21 - 3.21i)T - 17iT^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + (1.72 - 1.72i)T - 23iT^{2} \)
29 \( 1 + 8.70iT - 29T^{2} \)
31 \( 1 + 2.23iT - 31T^{2} \)
37 \( 1 + (4.21 - 4.21i)T - 37iT^{2} \)
41 \( 1 + 1.59T + 41T^{2} \)
43 \( 1 + (4.09 - 4.09i)T - 43iT^{2} \)
47 \( 1 + (0.691 + 0.691i)T + 47iT^{2} \)
53 \( 1 + (3.07 + 3.07i)T + 53iT^{2} \)
59 \( 1 - 3.06T + 59T^{2} \)
61 \( 1 + 3.44T + 61T^{2} \)
67 \( 1 + (4.51 + 4.51i)T + 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (-6.84 - 6.84i)T + 73iT^{2} \)
79 \( 1 + 0.973T + 79T^{2} \)
83 \( 1 + (-0.453 + 0.453i)T - 83iT^{2} \)
89 \( 1 - 4.23iT - 89T^{2} \)
97 \( 1 + (-7.14 + 7.14i)T - 97iT^{2} \)
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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

−12.09750564678114081663752787049, −11.33730367539888552506620928178, −9.897425308822187993135215382115, −8.822850224362638479310476345185, −8.283423179890846360680478641926, −7.75583483743621505364820571061, −6.11667864919122207599991818756, −5.08569246358643107240420504906, −2.45350545364561619093950087858, −1.65613977053951714189558617124, 1.98255276241600182583495254897, 3.42911804521163661870955677299, 4.73177467559011348929950656704, 6.86753902431564649805475067607, 7.43376770045736862604293991387, 8.627904554195719399823574697970, 9.540972197050607005236152036261, 10.42340688957783436198945029180, 10.91690435608967953768111848301, 11.95789803317719244699753721010

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