↑

Properties

Label 2-220-20.3-c1-0-12
Degree $2$
Conductor $220$
Sign $-0.0370 - 0.999i$
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  + (0.493 + 1.32i)2-s + (0.130 − 0.130i)3-s + (−1.51 + 1.30i)4-s + (2.21 + 0.272i)5-s + (0.238 + 0.108i)6-s + (0.450 + 0.450i)7-s + (−2.48 − 1.35i)8-s + 2.96i·9-s + (0.733 + 3.07i)10-s − i·11-s + (−0.0268 + 0.369i)12-s + (1.30 + 1.30i)13-s + (−0.374 + 0.818i)14-s + (0.326 − 0.254i)15-s + (0.577 − 3.95i)16-s + (1.51 − 1.51i)17-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s + (0.0756 − 0.0756i)3-s + (−0.756 + 0.654i)4-s + (0.992 + 0.121i)5-s + (0.0972 + 0.0444i)6-s + (0.170 + 0.170i)7-s + (−0.876 − 0.480i)8-s + 0.988i·9-s + (0.232 + 0.972i)10-s − 0.301i·11-s + (−0.00774 + 0.106i)12-s + (0.362 + 0.362i)13-s + (−0.100 + 0.218i)14-s + (0.0842 − 0.0658i)15-s + (0.144 − 0.989i)16-s + (0.366 − 0.366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0370 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0370 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.0370 - 0.999i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.0370 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07440 + 1.11500i\)
\(L(\frac12)\) \(\approx\) \(1.07440 + 1.11500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.493 - 1.32i)T \)
5 \( 1 + (-2.21 - 0.272i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-0.130 + 0.130i)T - 3iT^{2} \)
7 \( 1 + (-0.450 - 0.450i)T + 7iT^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + 13iT^{2} \)
17 \( 1 + (-1.51 + 1.51i)T - 17iT^{2} \)
19 \( 1 + 4.82T + 19T^{2} \)
23 \( 1 + (1.60 - 1.60i)T - 23iT^{2} \)
29 \( 1 + 6.86iT - 29T^{2} \)
31 \( 1 + 8.23iT - 31T^{2} \)
37 \( 1 + (0.0879 - 0.0879i)T - 37iT^{2} \)
41 \( 1 + 2.87T + 41T^{2} \)
43 \( 1 + (-7.00 + 7.00i)T - 43iT^{2} \)
47 \( 1 + (-5.17 - 5.17i)T + 47iT^{2} \)
53 \( 1 + (-0.125 - 0.125i)T + 53iT^{2} \)
59 \( 1 + 1.18T + 59T^{2} \)
61 \( 1 + 7.84T + 61T^{2} \)
67 \( 1 + (-3.17 - 3.17i)T + 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + (6.57 + 6.57i)T + 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (-7.91 + 7.91i)T - 83iT^{2} \)
89 \( 1 + 3.59iT - 89T^{2} \)
97 \( 1 + (8.41 - 8.41i)T - 97iT^{2} \)
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

−12.98649231701021033354422882791, −11.70028384221760536397739173219, −10.47799176469262821355596051289, −9.399203119021498618340306748165, −8.437149612174143841904577192829, −7.46235591450012273176703690542, −6.22077225805741762072022334487, −5.48898294629950833560653808902, −4.22503484356026165439368946955, −2.38791858444062734076472570180, 1.46464306812227999695390832575, 3.04628989110785480718491617064, 4.38640158234240047545632746202, 5.64378882631317730202009861559, 6.60767504745381473593545717893, 8.527432371715381820595606637291, 9.286875332641233300121794540324, 10.29005358584443503822442916768, 10.88433891485301656940713570815, 12.40390806143311928598282934698

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