Properties

Label 2-160-40.29-c5-0-4
Degree $2$
Conductor $160$
Sign $-0.850 - 0.525i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·3-s + (51.3 − 21.9i)5-s + 170. i·7-s − 241.·9-s + 39.8i·11-s − 537.·13-s + (−66.7 + 28.5i)15-s − 1.35e3i·17-s + 1.03e3i·19-s − 221. i·21-s + 1.61e3i·23-s + (2.15e3 − 2.26e3i)25-s + 628.·27-s + 5.91e3i·29-s − 9.61e3·31-s + ⋯
L(s)  = 1  − 0.0832·3-s + (0.919 − 0.393i)5-s + 1.31i·7-s − 0.993·9-s + 0.0992i·11-s − 0.881·13-s + (−0.0765 + 0.0327i)15-s − 1.13i·17-s + 0.660i·19-s − 0.109i·21-s + 0.634i·23-s + (0.690 − 0.723i)25-s + 0.165·27-s + 1.30i·29-s − 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7296728920\)
\(L(\frac12)\) \(\approx\) \(0.7296728920\)
\(L(\frac{7}{2})\) 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 \( 1 + (-51.3 + 21.9i)T \)
good3 \( 1 + 1.29T + 243T^{2} \)
7 \( 1 - 170. iT - 1.68e4T^{2} \)
11 \( 1 - 39.8iT - 1.61e5T^{2} \)
13 \( 1 + 537.T + 3.71e5T^{2} \)
17 \( 1 + 1.35e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.03e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.61e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.91e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.61e3T + 2.86e7T^{2} \)
37 \( 1 + 5.29e3T + 6.93e7T^{2} \)
41 \( 1 + 1.21e4T + 1.15e8T^{2} \)
43 \( 1 + 1.93e4T + 1.47e8T^{2} \)
47 \( 1 + 7.89e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.89e4T + 4.18e8T^{2} \)
59 \( 1 - 3.23e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.88e4iT - 8.44e8T^{2} \)
67 \( 1 - 882.T + 1.35e9T^{2} \)
71 \( 1 - 8.81e3T + 1.80e9T^{2} \)
73 \( 1 - 1.43e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.98e4T + 3.07e9T^{2} \)
83 \( 1 + 1.39e4T + 3.93e9T^{2} \)
89 \( 1 - 7.58e3T + 5.58e9T^{2} \)
97 \( 1 + 6.67e4iT - 8.58e9T^{2} \)
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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

−12.24473652942729295365425270294, −11.66733334710921045247161097694, −10.23632989609915443363494059309, −9.194115964220266248860474568468, −8.621785083229289463846832925579, −7.03875412203118045406556354259, −5.55427562391460347256672298794, −5.25226810152378232039541741004, −2.99037697527705419755462988487, −1.86647327710556194740603461641, 0.21935995984750357539039006393, 2.00859024980388393237937074465, 3.47755416664599261128187447562, 5.01261478751072933181800038925, 6.23923438753302818872657682822, 7.19735071882409336894113246274, 8.470871603653177034816009518986, 9.742200492485610806814087919130, 10.53824513401560075907304533886, 11.34012286821858504977005055089

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