↑

Properties

Label 2-160-20.3-c1-0-1
Degree $2$
Conductor $160$
Sign $0.850 - 0.525i$
Analytic cond. $1.27760$
Root an. cond. $1.13031$
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  + (−1 + i)3-s + (1 − 2i)5-s + (3 + 3i)7-s + i·9-s − 2i·11-s + (3 + 3i)13-s + (1 + 3i)15-s + (1 − i)17-s − 4·19-s − 6·21-s + (1 − i)23-s + (−3 − 4i)25-s + (−4 − 4i)27-s − 10i·31-s + (2 + 2i)33-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (0.447 − 0.894i)5-s + (1.13 + 1.13i)7-s + 0.333i·9-s − 0.603i·11-s + (0.832 + 0.832i)13-s + (0.258 + 0.774i)15-s + (0.242 − 0.242i)17-s − 0.917·19-s − 1.30·21-s + (0.208 − 0.208i)23-s + (−0.600 − 0.800i)25-s + (−0.769 − 0.769i)27-s − 1.79i·31-s + (0.348 + 0.348i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07266 + 0.304722i\)
\(L(\frac12)\) \(\approx\) \(1.07266 + 0.304722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + (-3 - 3i)T + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (5 - 5i)T - 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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

−12.95925770827375997058743013906, −11.63382711835295488583924754104, −11.28655009197773379600839819346, −9.945546337542022813772769662669, −8.772467860817053579338485519718, −8.206353204226717108679830505973, −6.12815307755934488180846606296, −5.28864812120296300610821443940, −4.38004981953623604220263144997, −1.95726066495511050852451073507, 1.51748122480919915396563750512, 3.65475271465668673728337717689, 5.28133571641083190207100039484, 6.58323582471142846291919349073, 7.28484508312509426651299023159, 8.488576150638774646669330552913, 10.22712027625269440624422582566, 10.74819187576907318581233985160, 11.69293023331710011895744125099, 12.84139341637347290761539981621

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