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Properties

Label 2-160-160.69-c1-0-10
Degree $2$
Conductor $160$
Sign $0.921 - 0.387i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.0348i)2-s + (−0.670 + 0.277i)3-s + (1.99 − 0.0985i)4-s + (0.174 + 2.22i)5-s + (−0.938 + 0.416i)6-s + (−0.00242 + 0.00242i)7-s + (2.82 − 0.209i)8-s + (−1.74 + 1.74i)9-s + (0.325 + 3.14i)10-s + (2.39 − 5.78i)11-s + (−1.31 + 0.620i)12-s + (−0.904 − 2.18i)13-s + (−0.00334 + 0.00351i)14-s + (−0.736 − 1.44i)15-s + (3.98 − 0.393i)16-s − 4.91·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0246i)2-s + (−0.387 + 0.160i)3-s + (0.998 − 0.0492i)4-s + (0.0782 + 0.996i)5-s + (−0.383 + 0.169i)6-s + (−0.000916 + 0.000916i)7-s + (0.997 − 0.0739i)8-s + (−0.582 + 0.582i)9-s + (0.102 + 0.994i)10-s + (0.722 − 1.74i)11-s + (−0.378 + 0.179i)12-s + (−0.250 − 0.605i)13-s + (−0.000893 + 0.000938i)14-s + (−0.190 − 0.373i)15-s + (0.995 − 0.0984i)16-s − 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72819 + 0.348500i\)
\(L(\frac12)\) \(\approx\) \(1.72819 + 0.348500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0348i)T \)
5 \( 1 + (-0.174 - 2.22i)T \)
good3 \( 1 + (0.670 - 0.277i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.00242 - 0.00242i)T - 7iT^{2} \)
11 \( 1 + (-2.39 + 5.78i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.904 + 2.18i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.91T + 17T^{2} \)
19 \( 1 + (1.34 - 0.559i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.03 - 1.03i)T + 23iT^{2} \)
29 \( 1 + (-1.65 - 4.00i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 7.82T + 31T^{2} \)
37 \( 1 + (-3.06 + 7.38i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.0199 - 0.0199i)T - 41iT^{2} \)
43 \( 1 + (6.09 + 2.52i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.78T + 47T^{2} \)
53 \( 1 + (-8.86 - 3.67i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.32 + 1.78i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.84 - 6.87i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.87 - 2.43i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.33 - 1.33i)T + 71iT^{2} \)
73 \( 1 + (-3.25 - 3.25i)T + 73iT^{2} \)
79 \( 1 - 7.56iT - 79T^{2} \)
83 \( 1 + (3.50 + 8.45i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.60 + 3.60i)T + 89iT^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−13.15485317672325649469027262035, −11.79383552237507730052347828630, −10.95182910443472063124010277953, −10.65829266135410891613560988005, −8.779055423775151127030305997844, −7.37189680414643991926253824194, −6.21169737851318076090104061641, −5.49403678419953838578342544953, −3.83015353236004157286793696497, −2.62909057116220130475133239891, 1.97782626192591275603343563341, 4.13829432281055862089707140226, 4.95159494770990106339098624638, 6.30359330335582041307929654322, 7.15373605398111531905645330000, 8.760527236083696557480128670392, 9.813057456475986714822035061129, 11.34204171945821047166256659885, 12.05923577148504119361388429953, 12.69300671734825437204586395978

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