Properties

Label 2-160-5.4-c3-0-9
Degree $2$
Conductor $160$
Sign $0.640 - 0.767i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.30i·3-s + (8.58 + 7.16i)5-s − 28.3i·7-s + 8.49·9-s + 65.2·11-s + 33.6i·13-s + (−30.8 + 36.9i)15-s + 73.3i·17-s − 134.·19-s + 121.·21-s − 14.7i·23-s + (22.3 + 122. i)25-s + 152. i·27-s + 224.·29-s + 68.8·31-s + ⋯
L(s)  = 1  + 0.827i·3-s + (0.767 + 0.640i)5-s − 1.52i·7-s + 0.314·9-s + 1.78·11-s + 0.718i·13-s + (−0.530 + 0.635i)15-s + 1.04i·17-s − 1.61·19-s + 1.26·21-s − 0.133i·23-s + (0.178 + 0.983i)25-s + 1.08i·27-s + 1.43·29-s + 0.398·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.640 - 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.90304 + 0.890302i\)
\(L(\frac12)\) \(\approx\) \(1.90304 + 0.890302i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.58 - 7.16i)T \)
good3 \( 1 - 4.30iT - 27T^{2} \)
7 \( 1 + 28.3iT - 343T^{2} \)
11 \( 1 - 65.2T + 1.33e3T^{2} \)
13 \( 1 - 33.6iT - 2.19e3T^{2} \)
17 \( 1 - 73.3iT - 4.91e3T^{2} \)
19 \( 1 + 134.T + 6.85e3T^{2} \)
23 \( 1 + 14.7iT - 1.21e4T^{2} \)
29 \( 1 - 224.T + 2.43e4T^{2} \)
31 \( 1 - 68.8T + 2.97e4T^{2} \)
37 \( 1 + 196. iT - 5.06e4T^{2} \)
41 \( 1 + 143.T + 6.89e4T^{2} \)
43 \( 1 - 15.0iT - 7.95e4T^{2} \)
47 \( 1 + 134. iT - 1.03e5T^{2} \)
53 \( 1 + 262. iT - 1.48e5T^{2} \)
59 \( 1 - 119.T + 2.05e5T^{2} \)
61 \( 1 - 16.5T + 2.26e5T^{2} \)
67 \( 1 + 545. iT - 3.00e5T^{2} \)
71 \( 1 + 199.T + 3.57e5T^{2} \)
73 \( 1 + 43.2iT - 3.89e5T^{2} \)
79 \( 1 + 438.T + 4.93e5T^{2} \)
83 \( 1 - 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 + 723.T + 7.04e5T^{2} \)
97 \( 1 + 1.13e3iT - 9.12e5T^{2} \)
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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

−12.65102516625626719740727689626, −11.18540610345079145280014272559, −10.42455702716189583744869285012, −9.807126889867510990824131643389, −8.689632895719970032627158829019, −6.91814675587438902916009517372, −6.41569996542694090543482118312, −4.38488680841972823627493930141, −3.79901246545787289774327788112, −1.56193145737809749519729641611, 1.23966192492169492694470605436, 2.48600369849055589303356204426, 4.64008319903868148717623633521, 6.01327237379624715361047431653, 6.68896330492514016541639423111, 8.374677045455460130359117812571, 9.037935095013584841341932593067, 10.05150361930136669543305855544, 11.78719477915906677016778557224, 12.27462208942282877977934992218

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