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Properties

Label 2-220-55.43-c1-0-4
Degree $2$
Conductor $220$
Sign $-0.850 + 0.525i$
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 − i)3-s + (−1 + 2i)5-s + (−3.31 − 3.31i)7-s − i·9-s − 3.31i·11-s + (−3.31 + 3.31i)13-s + (3 − i)15-s + (−3.31 − 3.31i)17-s + 6.63i·21-s + (3 + 3i)23-s + (−3 − 4i)25-s + (−4 + 4i)27-s + 6.63·29-s − 4·31-s + (−3.31 + 3.31i)33-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−0.447 + 0.894i)5-s + (−1.25 − 1.25i)7-s − 0.333i·9-s − 1.00i·11-s + (−0.919 + 0.919i)13-s + (0.774 − 0.258i)15-s + (−0.804 − 0.804i)17-s + 1.44i·21-s + (0.625 + 0.625i)23-s + (−0.600 − 0.800i)25-s + (−0.769 + 0.769i)27-s + 1.23·29-s − 0.718·31-s + (−0.577 + 0.577i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115523 - 0.406661i\)
\(L(\frac12)\) \(\approx\) \(0.115523 - 0.406661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1 - 2i)T \)
11 \( 1 + 3.31iT \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (3.31 + 3.31i)T + 7iT^{2} \)
13 \( 1 + (3.31 - 3.31i)T - 13iT^{2} \)
17 \( 1 + (3.31 + 3.31i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-3 - 3i)T + 23iT^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-3.31 + 3.31i)T - 43iT^{2} \)
47 \( 1 + (-5 + 5i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (3.31 - 3.31i)T - 73iT^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (-3.31 + 3.31i)T - 83iT^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + (-5 + 5i)T - 97iT^{2} \)
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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

−11.74775462195726139963478977592, −11.09811115715863595260463233186, −10.08142101359487118905708034231, −9.080618950628470517460730948177, −7.20416048894738857540437318043, −7.03035059618103168521804987436, −6.02690240422081118497000717493, −4.14074196460287074107750495619, −3.00368105790245311833605926492, −0.36031884962829309962385165188, 2.62926123939872058810029953376, 4.42257670886515236174433347081, 5.25756665434140280463819636959, 6.33727932095723965907224023997, 7.80572327821513110434623417221, 8.953867326704648180682716666181, 9.747452450019939176352717903471, 10.65714131791602733057834674704, 11.95893382003380903826260331267, 12.59423949365843407189370766262

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