↑

Properties

Label 2-220-55.49-c1-0-4
Degree $2$
Conductor $220$
Sign $0.560 + 0.828i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.447 − 0.616i)3-s + (1.93 − 1.11i)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 + (0.178 − 1.69i)15-s + (1.74 + 0.567i)17-s + (3.50 + 2.54i)19-s − 3.66·21-s − 0.109i·23-s + (2.50 − 4.33i)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 + (0.866 − 0.499i)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 + (0.0459 − 0.437i)15-s + (0.423 + 0.137i)17-s + (0.804 + 0.584i)19-s − 0.799·21-s − 0.0227i·23-s + (0.500 − 0.866i)25-s + (0.755 + 0.245i)27-s + (−0.786 + 0.571i)29-s + (−0.0647 − 0.199i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.560 + 0.828i$
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: \((2,\ 220,\ (\ :1/2),\ 0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22278 - 0.648783i\)
\(L(\frac12)\) \(\approx\) \(1.22278 - 0.648783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 1.11i)T \)
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 \( 1 + (2.20 - 0.715i)T + (78.4 - 57.0i)T^{2} \)
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

−12.45782962285184555734256799063, −11.03957466744512723559204142976, −9.960820499443048413537303343574, −9.488902360506961423304194428588, −8.036812814244111932338675078335, −7.07104467458843884962717775066, −6.08583550575756052066039554430, −4.65643164348773527162452224871, −3.23644307589015217064685941529, −1.35268210146184861980311027124, 2.37103800279450939905545809700, 3.46613425306442714402737556380, 5.27919416708347325612162755325, 6.26700134198688120844437750704, 7.16855357145166522800925219314, 9.003378418671442838851863973795, 9.525602439281300890093971735062, 10.04539944927265320904322430075, 11.68929877707424878824311759607, 12.40310517631997220825914153064

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