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Properties

Label 2-220-44.19-c1-0-4
Degree $2$
Conductor $220$
Sign $0.998 + 0.0519i$
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  + (−0.117 − 1.40i)2-s + (−3.21 + 1.04i)3-s + (−1.97 + 0.331i)4-s + (−0.809 − 0.587i)5-s + (1.85 + 4.40i)6-s + (0.161 − 0.498i)7-s + (0.699 + 2.74i)8-s + (6.82 − 4.95i)9-s + (−0.733 + 1.20i)10-s + (2.73 + 1.87i)11-s + (5.99 − 3.12i)12-s + (2.22 + 3.06i)13-s + (−0.720 − 0.169i)14-s + (3.21 + 1.04i)15-s + (3.77 − 1.30i)16-s + (−1.63 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.0832 − 0.996i)2-s + (−1.85 + 0.603i)3-s + (−0.986 + 0.165i)4-s + (−0.361 − 0.262i)5-s + (0.755 + 1.79i)6-s + (0.0611 − 0.188i)7-s + (0.247 + 0.968i)8-s + (2.27 − 1.65i)9-s + (−0.231 + 0.382i)10-s + (0.824 + 0.566i)11-s + (1.73 − 0.902i)12-s + (0.617 + 0.850i)13-s + (−0.192 − 0.0452i)14-s + (0.830 + 0.269i)15-s + (0.944 − 0.327i)16-s + (−0.396 + 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.998 + 0.0519i$
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: \((2,\ 220,\ (\ :1/2),\ 0.998 + 0.0519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545584 - 0.0141696i\)
\(L(\frac12)\) \(\approx\) \(0.545584 - 0.0141696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.117 + 1.40i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-2.73 - 1.87i)T \)
good3 \( 1 + (3.21 - 1.04i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.161 + 0.498i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.22 - 3.06i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.63 - 2.25i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.531 - 1.63i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.45iT - 23T^{2} \)
29 \( 1 + (-0.474 - 0.154i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.66 - 3.67i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.02 - 6.22i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.86 + 1.25i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.72T + 43T^{2} \)
47 \( 1 + (-6.44 + 2.09i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.379 - 0.275i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.529 - 0.171i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.95 - 8.19i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.46iT - 67T^{2} \)
71 \( 1 + (-0.730 + 1.00i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.0 - 3.59i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.67 - 7.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.87 + 2.81i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.48T + 89T^{2} \)
97 \( 1 + (6.27 - 4.55i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.20345696734604545934112658212, −11.24784986971566703790743493647, −10.66474653197776598266150460673, −9.742302930679247548299818972177, −8.765778317532832774133368689802, −6.95876354563406317205406832482, −5.83917520223754224398247852484, −4.47284413233946156202622883800, −4.07479864383826568586626514992, −1.21566058457944414330949118946, 0.76432794432755225004382625438, 4.15526623749758918471848145352, 5.44987397624670148384597903742, 6.09034848005571224108906793457, 7.03091236259624572570681691002, 7.85305637370945157706109229257, 9.294422306003187781506311968479, 10.62811121830001015251533910620, 11.33990253877818077436958223463, 12.27381040993286656814231793140

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