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Properties

Label 2-220-44.35-c1-0-7
Degree $2$
Conductor $220$
Sign $0.753 - 0.657i$
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  + (−0.478 + 1.33i)2-s + (−0.295 − 0.406i)3-s + (−1.54 − 1.27i)4-s + (−0.309 − 0.951i)5-s + (0.682 − 0.198i)6-s + (2.59 + 1.88i)7-s + (2.43 − 1.44i)8-s + (0.848 − 2.61i)9-s + (1.41 + 0.0434i)10-s + (0.569 + 3.26i)11-s + (−0.0617 + 1.00i)12-s + (3.70 + 1.20i)13-s + (−3.75 + 2.55i)14-s + (−0.295 + 0.406i)15-s + (0.760 + 3.92i)16-s + (2.89 − 0.941i)17-s + ⋯
L(s)  = 1  + (−0.338 + 0.941i)2-s + (−0.170 − 0.234i)3-s + (−0.771 − 0.636i)4-s + (−0.138 − 0.425i)5-s + (0.278 − 0.0811i)6-s + (0.981 + 0.712i)7-s + (0.859 − 0.510i)8-s + (0.282 − 0.870i)9-s + (0.447 + 0.0137i)10-s + (0.171 + 0.985i)11-s + (−0.0178 + 0.289i)12-s + (1.02 + 0.333i)13-s + (−1.00 + 0.682i)14-s + (−0.0763 + 0.105i)15-s + (0.190 + 0.981i)16-s + (0.703 − 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970218 + 0.363708i\)
\(L(\frac12)\) \(\approx\) \(0.970218 + 0.363708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.478 - 1.33i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.569 - 3.26i)T \)
good3 \( 1 + (0.295 + 0.406i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.59 - 1.88i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-3.70 - 1.20i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.89 + 0.941i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.0572 + 0.0415i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.67iT - 23T^{2} \)
29 \( 1 + (2.43 - 3.34i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.894 + 0.290i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.15 - 5.19i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.98 + 6.86i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.496T + 43T^{2} \)
47 \( 1 + (2.38 + 3.27i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.43 + 7.50i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.38 - 11.5i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.696 + 0.226i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.155iT - 67T^{2} \)
71 \( 1 + (14.2 - 4.64i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.84 - 10.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.20 + 9.87i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.52 - 4.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (2.04 - 6.29i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.39670717281735169210069418959, −11.58539929593628776827239140473, −10.22355993422354797501475822247, −9.119066890887163134988865387502, −8.468115904510180995895131183046, −7.36303547617422283599852026680, −6.33368803664173613199846619573, −5.24006504827474464520680701025, −4.15640963185589922360045945893, −1.43421216875951091172694481654, 1.44448006909679859777449135051, 3.33007045979537256127008642505, 4.39633557479381308943891932670, 5.72679684830967359296217367668, 7.67683530830445000201274919615, 8.100395133964111263264371459543, 9.473516908975903099003364434287, 10.66322368429298031762180531782, 10.99170540529299662376947923675, 11.74203862722087284549997866772

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