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Properties

Label 2-220-20.3-c1-0-5
Degree $2$
Conductor $220$
Sign $0.976 - 0.213i$
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  + (−1.12 − 0.857i)2-s + (0.635 − 0.635i)3-s + (0.530 + 1.92i)4-s + (−1.96 + 1.07i)5-s + (−1.25 + 0.170i)6-s + (2.25 + 2.25i)7-s + (1.05 − 2.62i)8-s + 2.19i·9-s + (3.12 + 0.475i)10-s − i·11-s + (1.56 + 0.887i)12-s + (3.81 + 3.81i)13-s + (−0.603 − 4.46i)14-s + (−0.565 + 1.92i)15-s + (−3.43 + 2.04i)16-s + (2.04 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.795 − 0.606i)2-s + (0.366 − 0.366i)3-s + (0.265 + 0.964i)4-s + (−0.877 + 0.479i)5-s + (−0.513 + 0.0694i)6-s + (0.851 + 0.851i)7-s + (0.373 − 0.927i)8-s + 0.731i·9-s + (0.988 + 0.150i)10-s − 0.301i·11-s + (0.450 + 0.256i)12-s + (1.05 + 1.05i)13-s + (−0.161 − 1.19i)14-s + (−0.145 + 0.497i)15-s + (−0.859 + 0.511i)16-s + (0.495 − 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.976 - 0.213i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.976 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886623 + 0.0957328i\)
\(L(\frac12)\) \(\approx\) \(0.886623 + 0.0957328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.857i)T \)
5 \( 1 + (1.96 - 1.07i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-0.635 + 0.635i)T - 3iT^{2} \)
7 \( 1 + (-2.25 - 2.25i)T + 7iT^{2} \)
13 \( 1 + (-3.81 - 3.81i)T + 13iT^{2} \)
17 \( 1 + (-2.04 + 2.04i)T - 17iT^{2} \)
19 \( 1 + 3.69T + 19T^{2} \)
23 \( 1 + (-2.52 + 2.52i)T - 23iT^{2} \)
29 \( 1 - 8.82iT - 29T^{2} \)
31 \( 1 + 6.21iT - 31T^{2} \)
37 \( 1 + (-1.82 + 1.82i)T - 37iT^{2} \)
41 \( 1 + 0.00765T + 41T^{2} \)
43 \( 1 + (5.73 - 5.73i)T - 43iT^{2} \)
47 \( 1 + (-1.19 - 1.19i)T + 47iT^{2} \)
53 \( 1 + (5.59 + 5.59i)T + 53iT^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 2.14T + 61T^{2} \)
67 \( 1 + (3.37 + 3.37i)T + 67iT^{2} \)
71 \( 1 + 0.207iT - 71T^{2} \)
73 \( 1 + (0.0410 + 0.0410i)T + 73iT^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-11.8 + 11.8i)T - 83iT^{2} \)
89 \( 1 - 5.49iT - 89T^{2} \)
97 \( 1 + (-8.27 + 8.27i)T - 97iT^{2} \)
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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

−12.02248588291314398797118836801, −11.19880688046358960369235286897, −10.76188391536327613277297343353, −9.098041084073446700339313952413, −8.398126205458740455663616431974, −7.70795912472549661335245034077, −6.56408103595674591112343276201, −4.61341756862519114578008087879, −3.14361094563429899058703344062, −1.83474845829345656917256561361, 1.06490985676965274702118492390, 3.67716448612102021369433152199, 4.83341439626734801106502094567, 6.27953215998650740132984360598, 7.62879238016649699543432616368, 8.215303808750242256033883211758, 9.049915044097271451614702847427, 10.29511173174802468267905709241, 10.97157635053633939103320843745, 12.06024905126489751375539840843

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