Properties

Label 2-160-8.5-c1-0-3
Degree $2$
Conductor $160$
Sign $-0.258 + 0.965i$
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  − 2.73i·3-s i·5-s − 0.732·7-s − 4.46·9-s + 2i·11-s − 3.46i·13-s − 2.73·15-s + 3.46·17-s − 0.535i·19-s + 2i·21-s + 6.19·23-s − 25-s + 3.99i·27-s + 6.92i·29-s + 5.46·31-s + ⋯
L(s)  = 1  − 1.57i·3-s − 0.447i·5-s − 0.276·7-s − 1.48·9-s + 0.603i·11-s − 0.960i·13-s − 0.705·15-s + 0.840·17-s − 0.122i·19-s + 0.436i·21-s + 1.29·23-s − 0.200·25-s + 0.769i·27-s + 1.28i·29-s + 0.981·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658367 - 0.858000i\)
\(L(\frac12)\) \(\approx\) \(0.658367 - 0.858000i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + 2.73iT - 3T^{2} \)
7 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 0.535iT - 19T^{2} \)
23 \( 1 - 6.19T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 7.46iT - 59T^{2} \)
61 \( 1 + 8.92iT - 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 + 1.26iT - 83T^{2} \)
89 \( 1 - 8.92T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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

−12.66957005593011508255180528453, −12.04759357181650432074593686855, −10.73654620022073340098703779697, −9.387370989585894219543836341344, −8.163922623914325870223683387448, −7.37910142044265702757719372060, −6.34632751921245849871945507548, −5.09947870171537809908534329092, −2.94655939220162838838456849961, −1.19825709651491645034593071036, 3.06159837411850381180474499274, 4.15431509374419544021751980963, 5.38663634296394194511571409250, 6.66801448540019197078717706831, 8.288051536299278320481287814306, 9.410217297651719987564535040589, 10.06797603690172882670540691445, 11.05570888154484132433956799558, 11.81418333633619723700614308237, 13.39013680020442365032820602602

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