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Properties

Label 2-220-44.19-c1-0-14
Degree $2$
Conductor $220$
Sign $0.746 - 0.665i$
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  + (0.893 + 1.09i)2-s + (2.27 − 0.737i)3-s + (−0.402 + 1.95i)4-s + (−0.809 − 0.587i)5-s + (2.83 + 1.82i)6-s + (0.461 − 1.42i)7-s + (−2.50 + 1.30i)8-s + (2.18 − 1.58i)9-s + (−0.0788 − 1.41i)10-s + (1.72 + 2.83i)11-s + (0.531 + 4.74i)12-s + (−1.61 − 2.22i)13-s + (1.97 − 0.764i)14-s + (−2.27 − 0.737i)15-s + (−3.67 − 1.57i)16-s + (−3.56 + 4.90i)17-s + ⋯
L(s)  = 1  + (0.631 + 0.774i)2-s + (1.31 − 0.425i)3-s + (−0.201 + 0.979i)4-s + (−0.361 − 0.262i)5-s + (1.15 + 0.746i)6-s + (0.174 − 0.537i)7-s + (−0.886 + 0.463i)8-s + (0.728 − 0.529i)9-s + (−0.0249 − 0.446i)10-s + (0.519 + 0.854i)11-s + (0.153 + 1.36i)12-s + (−0.447 − 0.616i)13-s + (0.526 − 0.204i)14-s + (−0.586 − 0.190i)15-s + (−0.919 − 0.394i)16-s + (−0.863 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01851 + 0.768486i\)
\(L(\frac12)\) \(\approx\) \(2.01851 + 0.768486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.893 - 1.09i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-1.72 - 2.83i)T \)
good3 \( 1 + (-2.27 + 0.737i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.461 + 1.42i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.61 + 2.22i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.56 - 4.90i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.25 + 3.85i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.20iT - 23T^{2} \)
29 \( 1 + (-1.78 - 0.578i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.545 + 0.751i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.08 - 6.40i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.96 - 1.28i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.04T + 43T^{2} \)
47 \( 1 + (-8.50 + 2.76i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.02 + 4.37i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-11.1 - 3.63i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.84 + 2.53i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.6iT - 67T^{2} \)
71 \( 1 + (7.69 - 10.5i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (7.42 + 2.41i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.96 - 2.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.82 - 2.05i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.08T + 89T^{2} \)
97 \( 1 + (-12.0 + 8.75i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.87250293402601501564922460819, −11.86336410381016509867990708171, −10.35096565558050644678209629313, −8.853893103958396148266818526512, −8.375325644023398136072169008532, −7.35404997138492164983875170470, −6.60823978476092676391294750357, −4.77397112409260330968221995088, −3.85168176640926824190319936368, −2.41866844405229758441029428160, 2.19770431297163853221125012536, 3.32130574402360206839307107485, 4.22796137827155187892698610369, 5.65932232363909633408555937214, 7.16747016770166219517407273424, 8.675812665762617874096793976444, 9.187161928373647135536523553349, 10.20325136289352966408789711045, 11.46930108813690124170895233197, 11.96029278129249837027573865131

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