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Properties

Label 2-220-20.7-c1-0-7
Degree $2$
Conductor $220$
Sign $-0.426 - 0.904i$
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  + (0.513 + 1.31i)2-s + (0.912 + 0.912i)3-s + (−1.47 + 1.35i)4-s + (1.46 + 1.68i)5-s + (−0.733 + 1.67i)6-s + (0.0653 − 0.0653i)7-s + (−2.54 − 1.24i)8-s − 1.33i·9-s + (−1.47 + 2.79i)10-s − i·11-s + (−2.57 − 0.107i)12-s + (0.473 − 0.473i)13-s + (0.119 + 0.0524i)14-s + (−0.204 + 2.87i)15-s + (0.332 − 3.98i)16-s + (−1.01 − 1.01i)17-s + ⋯
L(s)  = 1  + (0.363 + 0.931i)2-s + (0.526 + 0.526i)3-s + (−0.735 + 0.677i)4-s + (0.655 + 0.755i)5-s + (−0.299 + 0.682i)6-s + (0.0246 − 0.0246i)7-s + (−0.898 − 0.439i)8-s − 0.444i·9-s + (−0.465 + 0.884i)10-s − 0.301i·11-s + (−0.744 − 0.0309i)12-s + (0.131 − 0.131i)13-s + (0.0319 + 0.0140i)14-s + (−0.0528 + 0.743i)15-s + (0.0830 − 0.996i)16-s + (−0.245 − 0.245i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.426 - 0.904i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895376 + 1.41263i\)
\(L(\frac12)\) \(\approx\) \(0.895376 + 1.41263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.513 - 1.31i)T \)
5 \( 1 + (-1.46 - 1.68i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-0.912 - 0.912i)T + 3iT^{2} \)
7 \( 1 + (-0.0653 + 0.0653i)T - 7iT^{2} \)
13 \( 1 + (-0.473 + 0.473i)T - 13iT^{2} \)
17 \( 1 + (1.01 + 1.01i)T + 17iT^{2} \)
19 \( 1 + 2.65T + 19T^{2} \)
23 \( 1 + (-5.32 - 5.32i)T + 23iT^{2} \)
29 \( 1 + 5.94iT - 29T^{2} \)
31 \( 1 - 1.44iT - 31T^{2} \)
37 \( 1 + (5.27 + 5.27i)T + 37iT^{2} \)
41 \( 1 + 5.13T + 41T^{2} \)
43 \( 1 + (-4.06 - 4.06i)T + 43iT^{2} \)
47 \( 1 + (-7.89 + 7.89i)T - 47iT^{2} \)
53 \( 1 + (-2.60 + 2.60i)T - 53iT^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 7.06T + 61T^{2} \)
67 \( 1 + (5.47 - 5.47i)T - 67iT^{2} \)
71 \( 1 + 1.80iT - 71T^{2} \)
73 \( 1 + (-5.16 + 5.16i)T - 73iT^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 - 5.69iT - 89T^{2} \)
97 \( 1 + (0.398 + 0.398i)T + 97iT^{2} \)
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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

−12.99528956806793907387481975741, −11.69929868977727833621008653415, −10.44655519028256394176266677415, −9.404779455738360843705232214970, −8.744727081624676930662085895388, −7.41660595154725703310755750736, −6.45818953842696301791973665863, −5.45689987737496628876318504006, −3.99094817594212426516128630014, −2.92349695545325318497426535473, 1.53826453368582009010875250432, 2.67895101940542786071787719776, 4.42753997541525481587961987651, 5.39877916381151534519686432897, 6.79809656778005612848940669327, 8.458546605466202034299575213670, 8.947556090944977247574785811283, 10.18802176646334227854863240184, 10.96850244070389272999125904505, 12.36971277368615690434681009611

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