↑

Properties

Label 2-220-4.3-c2-0-4
Degree $2$
Conductor $220$
Sign $-0.272 - 0.962i$
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  + (−0.274 − 1.98i)2-s + 4.52i·3-s + (−3.84 + 1.08i)4-s + 2.23·5-s + (8.96 − 1.24i)6-s + 1.58i·7-s + (3.21 + 7.32i)8-s − 11.5·9-s + (−0.614 − 4.42i)10-s + 3.31i·11-s + (−4.92 − 17.4i)12-s − 18.2·13-s + (3.13 − 0.435i)14-s + 10.1i·15-s + (13.6 − 8.37i)16-s − 13.8·17-s + ⋯
L(s)  = 1  + (−0.137 − 0.990i)2-s + 1.50i·3-s + (−0.962 + 0.272i)4-s + 0.447·5-s + (1.49 − 0.207i)6-s + 0.226i·7-s + (0.401 + 0.915i)8-s − 1.27·9-s + (−0.0614 − 0.442i)10-s + 0.301i·11-s + (−0.410 − 1.45i)12-s − 1.40·13-s + (0.224 − 0.0310i)14-s + 0.674i·15-s + (0.851 − 0.523i)16-s − 0.812·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(5.99456\)
Root analytic conductor: \(2.44838\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1),\ -0.272 - 0.962i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.560355 + 0.740728i\)
\(L(\frac12)\) \(\approx\) \(0.560355 + 0.740728i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.274 + 1.98i)T \)
5 \( 1 - 2.23T \)
11 \( 1 - 3.31iT \)
good3 \( 1 - 4.52iT - 9T^{2} \)
7 \( 1 - 1.58iT - 49T^{2} \)
13 \( 1 + 18.2T + 169T^{2} \)
17 \( 1 + 13.8T + 289T^{2} \)
19 \( 1 - 13.5iT - 361T^{2} \)
23 \( 1 - 34.1iT - 529T^{2} \)
29 \( 1 + 26.1T + 841T^{2} \)
31 \( 1 + 0.621iT - 961T^{2} \)
37 \( 1 - 9.06T + 1.36e3T^{2} \)
41 \( 1 - 42.3T + 1.68e3T^{2} \)
43 \( 1 + 30.1iT - 1.84e3T^{2} \)
47 \( 1 - 46.6iT - 2.20e3T^{2} \)
53 \( 1 - 23.6T + 2.80e3T^{2} \)
59 \( 1 + 88.7iT - 3.48e3T^{2} \)
61 \( 1 + 49.4T + 3.72e3T^{2} \)
67 \( 1 - 113. iT - 4.48e3T^{2} \)
71 \( 1 + 55.1iT - 5.04e3T^{2} \)
73 \( 1 - 123.T + 5.32e3T^{2} \)
79 \( 1 + 0.962iT - 6.24e3T^{2} \)
83 \( 1 + 29.6iT - 6.88e3T^{2} \)
89 \( 1 - 57.8T + 7.92e3T^{2} \)
97 \( 1 - 176.T + 9.40e3T^{2} \)
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

−12.10593033707531486751808310456, −11.15482660305551638630050681352, −10.30852787813709582729028037547, −9.536173041467879644053923389518, −9.118909497586796520655673450335, −7.62740354676652230694999969494, −5.57042785822574050414123430937, −4.70381782630483524903859654710, −3.64897944556010362159131349432, −2.25465511172402584606429050934, 0.52159366675445576672225233581, 2.34525410928129014421702066457, 4.63512502307635427719142864183, 5.93046829357846517725596740504, 6.84664042393251876099231655440, 7.45313528173887018331174366682, 8.511609903474244077840652835264, 9.485655237450933201638279510181, 10.76046335357918835823065072114, 12.19443227425325026789536697200

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