↑

Properties

Label 2-220-11.6-c2-0-6
Degree $2$
Conductor $220$
Sign $-0.972 + 0.233i$
Analytic cond. $5.99456$
Root an. cond. $2.44838$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.30 − 1.67i)3-s + (0.690 − 2.12i)5-s + (1.86 + 2.56i)7-s + (−0.274 − 0.844i)9-s + (−8.20 − 7.32i)11-s + (1.09 − 0.355i)13-s + (−5.15 + 3.74i)15-s + (−16.6 − 5.41i)17-s + (−11.7 + 16.2i)19-s − 9.01i·21-s − 36.9·23-s + (−4.04 − 2.93i)25-s + (−8.70 + 26.7i)27-s + (−2.94 − 4.04i)29-s + (−18.8 − 58.1i)31-s + ⋯
L(s)  = 1  + (−0.768 − 0.558i)3-s + (0.138 − 0.425i)5-s + (0.265 + 0.365i)7-s + (−0.0304 − 0.0938i)9-s + (−0.746 − 0.665i)11-s + (0.0842 − 0.0273i)13-s + (−0.343 + 0.249i)15-s + (−0.980 − 0.318i)17-s + (−0.619 + 0.852i)19-s − 0.429i·21-s − 1.60·23-s + (−0.161 − 0.117i)25-s + (−0.322 + 0.992i)27-s + (−0.101 − 0.139i)29-s + (−0.609 − 1.87i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.972 + 0.233i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0616145 - 0.519665i\)
\(L(\frac12)\) \(\approx\) \(0.0616145 - 0.519665i\)
\(L(2)\) 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 \( 1 + (-0.690 + 2.12i)T \)
11 \( 1 + (8.20 + 7.32i)T \)
good3 \( 1 + (2.30 + 1.67i)T + (2.78 + 8.55i)T^{2} \)
7 \( 1 + (-1.86 - 2.56i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-1.09 + 0.355i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (16.6 + 5.41i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (11.7 - 16.2i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 36.9T + 529T^{2} \)
29 \( 1 + (2.94 + 4.04i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (18.8 + 58.1i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-21.0 + 15.3i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-35.4 + 48.7i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 65.3iT - 1.84e3T^{2} \)
47 \( 1 + (-21.5 - 15.6i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-0.957 - 2.94i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-7.53 + 5.47i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-32.2 - 10.4i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 17.1T + 4.48e3T^{2} \)
71 \( 1 + (-15.6 + 48.2i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-24.2 - 33.4i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-40.1 + 13.0i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (76.7 + 24.9i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 109.T + 7.92e3T^{2} \)
97 \( 1 + (6.19 + 19.0i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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

−11.62124275099520265952246326054, −10.93537070157766438737122491418, −9.656614806556662843234681001697, −8.540174766216934750981280464186, −7.60037417085496421990951326675, −6.12498985584016253501535750507, −5.65967798692295421345379777659, −4.13340929230807037100050949784, −2.15667818007895407230486585987, −0.29613196059318324173255410984, 2.28383000673392033381122036026, 4.17514954444824312394322263785, 5.09618773095087372106309340497, 6.28291571252557014546467197414, 7.38746790092267431604493507460, 8.573204511407609116166596817754, 9.951156059085538697223626067120, 10.65269805486922043775384452168, 11.23555019603588665187453646331, 12.40408941377872597952216023296

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