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Properties

Label 2-160-20.3-c1-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)3-s + (−2 − i)5-s + (−2 − 2i)7-s − 5i·9-s + (−1 − i)13-s + (6 − 2i)15-s + (−5 + 5i)17-s − 4·19-s + 8·21-s + (−2 + 2i)23-s + (3 + 4i)25-s + (4 + 4i)27-s + 4i·29-s − 4i·31-s + (2 + 6i)35-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s + (−0.894 − 0.447i)5-s + (−0.755 − 0.755i)7-s − 1.66i·9-s + (−0.277 − 0.277i)13-s + (1.54 − 0.516i)15-s + (−1.21 + 1.21i)17-s − 0.917·19-s + 1.74·21-s + (−0.417 + 0.417i)23-s + (0.600 + 0.800i)25-s + (0.769 + 0.769i)27-s + 0.742i·29-s − 0.718i·31-s + (0.338 + 1.01i)35-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: \(1\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
good3 \( 1 + (2 - 2i)T - 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (5 - 5i)T - 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-6 + 6i)T - 43iT^{2} \)
47 \( 1 + (-2 - 2i)T + 47iT^{2} \)
53 \( 1 + (7 + 7i)T + 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + (-10 - 10i)T + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (-2 + 2i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.39120829607980716833016442920, −11.21213632369939348194190299961, −10.65330872072270715417064095239, −9.694235667765133118993808626927, −8.493258516651058848658419487236, −6.98204015859890003664557088210, −5.82427128957150840200781620625, −4.44760348718601702809622473482, −3.81985278896428350058783924405, 0, 2.55316889778586010739924050045, 4.62121038175294800783863441427, 6.17238974846732228512913187291, 6.76496165610320624134231613644, 7.79991666445786904202477652183, 9.155775512676237659889010969563, 10.72216362126657541072761406516, 11.53149773793812008891315253886, 12.24639074133331883143606070816

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