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Properties

Label 2-220-220.219-c1-0-12
Degree $2$
Conductor $220$
Sign $0.997 + 0.0750i$
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  + (−0.896 − 1.09i)2-s + 2.84·3-s + (−0.393 + 1.96i)4-s + (−1.64 + 1.51i)5-s + (−2.54 − 3.11i)6-s + 3.37i·7-s + (2.49 − 1.32i)8-s + 5.08·9-s + (3.13 + 0.449i)10-s + (1.92 − 2.70i)11-s + (−1.11 + 5.57i)12-s + 2.23·13-s + (3.69 − 3.02i)14-s + (−4.68 + 4.29i)15-s + (−3.69 − 1.54i)16-s + 2.76·17-s + ⋯
L(s)  = 1  + (−0.633 − 0.773i)2-s + 1.64·3-s + (−0.196 + 0.980i)4-s + (−0.737 + 0.675i)5-s + (−1.04 − 1.26i)6-s + 1.27i·7-s + (0.883 − 0.469i)8-s + 1.69·9-s + (0.989 + 0.142i)10-s + (0.580 − 0.814i)11-s + (−0.322 + 1.60i)12-s + 0.618·13-s + (0.986 − 0.808i)14-s + (−1.21 + 1.10i)15-s + (−0.922 − 0.385i)16-s + 0.670·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.997 + 0.0750i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.997 + 0.0750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34063 - 0.0504011i\)
\(L(\frac12)\) \(\approx\) \(1.34063 - 0.0504011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.896 + 1.09i)T \)
5 \( 1 + (1.64 - 1.51i)T \)
11 \( 1 + (-1.92 + 2.70i)T \)
good3 \( 1 - 2.84T + 3T^{2} \)
7 \( 1 - 3.37iT - 7T^{2} \)
13 \( 1 - 2.23T + 13T^{2} \)
17 \( 1 - 2.76T + 17T^{2} \)
19 \( 1 + 7.85T + 19T^{2} \)
23 \( 1 - 0.606T + 23T^{2} \)
29 \( 1 + 7.54iT - 29T^{2} \)
31 \( 1 - 2.02iT - 31T^{2} \)
37 \( 1 - 1.18iT - 37T^{2} \)
41 \( 1 + 2.04iT - 41T^{2} \)
43 \( 1 + 3.09iT - 43T^{2} \)
47 \( 1 + 8.76T + 47T^{2} \)
53 \( 1 + 4.21iT - 53T^{2} \)
59 \( 1 - 4.33iT - 59T^{2} \)
61 \( 1 - 2.84iT - 61T^{2} \)
67 \( 1 + 0.606T + 67T^{2} \)
71 \( 1 + 9.55iT - 71T^{2} \)
73 \( 1 - 6.00T + 73T^{2} \)
79 \( 1 + 3.84T + 79T^{2} \)
83 \( 1 - 8.39iT - 83T^{2} \)
89 \( 1 - 8.67T + 89T^{2} \)
97 \( 1 - 3.02iT - 97T^{2} \)
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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

−12.16059329638475988143896140354, −11.29466666962254840877195128375, −10.21104656800196933954365353555, −9.060980603681854522765763242184, −8.490249722086344046164095965007, −7.902569112762622516797644067631, −6.46492314923051080556676551242, −4.02762784608909246488957027160, −3.17331313025030955062920568372, −2.17865559169244424281991082495, 1.49039989090425321262231695610, 3.76556554282840678620146706888, 4.58692671501313617649581694727, 6.72025811013808639620047047820, 7.57680155006708483643416221867, 8.300777216681132939461984435943, 9.037422790840353500563982813780, 9.961443574900448728051167479139, 10.97223650459976155203458717544, 12.70564078736766401037377490047

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