↑

Properties

Label 2-220-5.4-c3-0-9
Degree $2$
Conductor $220$
Sign $-0.223 + 0.974i$
Analytic cond. $12.9804$
Root an. cond. $3.60283$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.35i·3-s + (−2.5 + 10.8i)5-s − 8.71i·7-s + 7.99·9-s − 11·11-s − 69.7i·13-s + (47.5 + 10.8i)15-s − 26.1i·17-s + 68·19-s − 38.0·21-s − 117. i·23-s + (−112. − 54.4i)25-s − 152. i·27-s − 260·29-s + 175·31-s + ⋯
L(s)  = 1  − 0.838i·3-s + (−0.223 + 0.974i)5-s − 0.470i·7-s + 0.296·9-s − 0.301·11-s − 1.48i·13-s + (0.817 + 0.187i)15-s − 0.373i·17-s + 0.821·19-s − 0.394·21-s − 1.06i·23-s + (−0.900 − 0.435i)25-s − 1.08i·27-s − 1.66·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(12.9804\)
Root analytic conductor: \(3.60283\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :3/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.888901 - 1.11592i\)
\(L(\frac12)\) \(\approx\) \(0.888901 - 1.11592i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 10.8i)T \)
11 \( 1 + 11T \)
good3 \( 1 + 4.35iT - 27T^{2} \)
7 \( 1 + 8.71iT - 343T^{2} \)
13 \( 1 + 69.7iT - 2.19e3T^{2} \)
17 \( 1 + 26.1iT - 4.91e3T^{2} \)
19 \( 1 - 68T + 6.85e3T^{2} \)
23 \( 1 + 117. iT - 1.21e4T^{2} \)
29 \( 1 + 260T + 2.43e4T^{2} \)
31 \( 1 - 175T + 2.97e4T^{2} \)
37 \( 1 + 169. iT - 5.06e4T^{2} \)
41 \( 1 + 380T + 6.89e4T^{2} \)
43 \( 1 + 305. iT - 7.95e4T^{2} \)
47 \( 1 - 305. iT - 1.03e5T^{2} \)
53 \( 1 + 453. iT - 1.48e5T^{2} \)
59 \( 1 - 143T + 2.05e5T^{2} \)
61 \( 1 - 676T + 2.26e5T^{2} \)
67 \( 1 - 527. iT - 3.00e5T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 331. iT - 3.89e5T^{2} \)
79 \( 1 + 218T + 4.93e5T^{2} \)
83 \( 1 - 758. iT - 5.71e5T^{2} \)
89 \( 1 + 1.27e3T + 7.04e5T^{2} \)
97 \( 1 - 771. iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.58639949252771081875024192292, −10.56376511680353018762772406797, −9.897713089129453427350787820338, −8.184744302757415598630857206314, −7.43236208204780404458438339723, −6.71616314121717887024652794718, −5.42010855150368838168576024934, −3.71069324037631876847393722610, −2.43516102729658480829965200191, −0.61625561810630477085879647291, 1.63036085856316864720049048689, 3.66923080155452296311962486922, 4.66051454224359381589732379373, 5.58230306706859417250584546049, 7.12897220904777658051349220430, 8.376785139584930230905656127960, 9.347860888262493976750365016387, 9.859277426808712231543834332675, 11.30284408933867571540769753388, 11.95074739280655888122812547081

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