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Properties

Label 2-220-55.49-c1-0-2
Degree $2$
Conductor $220$
Sign $0.882 - 0.471i$
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  + (−0.447 + 0.616i)3-s + (−0.909 − 2.04i)5-s + (2.82 + 3.89i)7-s + (0.747 + 2.30i)9-s + (2.43 − 2.25i)11-s + (2.78 − 0.903i)13-s + (1.66 + 0.354i)15-s + (−1.74 − 0.567i)17-s + (3.50 + 2.54i)19-s − 3.66·21-s + 0.109i·23-s + (−3.34 + 3.71i)25-s + (−3.92 − 1.27i)27-s + (−4.23 + 3.07i)29-s + (−0.360 − 1.11i)31-s + ⋯
L(s)  = 1  + (−0.258 + 0.355i)3-s + (−0.406 − 0.913i)5-s + (1.06 + 1.47i)7-s + (0.249 + 0.767i)9-s + (0.734 − 0.678i)11-s + (0.771 − 0.250i)13-s + (0.430 + 0.0914i)15-s + (−0.423 − 0.137i)17-s + (0.804 + 0.584i)19-s − 0.799·21-s + 0.0227i·23-s + (−0.669 + 0.743i)25-s + (−0.755 − 0.245i)27-s + (−0.786 + 0.571i)29-s + (−0.0647 − 0.199i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.882 - 0.471i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.882 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17983 + 0.295273i\)
\(L(\frac12)\) \(\approx\) \(1.17983 + 0.295273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.909 + 2.04i)T \)
11 \( 1 + (-2.43 + 2.25i)T \)
good3 \( 1 + (0.447 - 0.616i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-2.82 - 3.89i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2.78 + 0.903i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.74 + 0.567i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.50 - 2.54i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.109iT - 23T^{2} \)
29 \( 1 + (4.23 - 3.07i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.360 + 1.11i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.24 + 7.22i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.43 + 2.49i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.85iT - 43T^{2} \)
47 \( 1 + (-5.68 + 7.82i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.03 + 0.335i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.60 - 4.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.90 + 5.87i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 15.0iT - 67T^{2} \)
71 \( 1 + (1.20 - 3.69i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.51 + 11.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.67 + 5.16i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-8.31 - 2.70i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (-2.20 + 0.715i)T + (78.4 - 57.0i)T^{2} \)
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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

−12.06037108482543834657218867546, −11.56890330599569512676517123223, −10.64611917961240408572983225800, −9.053031333368107120250485956489, −8.645654218260044489117870658333, −7.60085322245781370671174096028, −5.68865856664249959714183080833, −5.19759724378446616085199818272, −3.85813867138592111172210376065, −1.76960236969263711431878316939, 1.36881735696049216745018816103, 3.61850725717956795661867382373, 4.52278518627554060367372905536, 6.42383314196300468146224785826, 7.09932088589820855327424858357, 7.901297795092422086318198375266, 9.383347674064829351019216137981, 10.51114899976072739030827787512, 11.32718137446433384688928475687, 11.89362677501710010312613561948

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