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Properties

Label 2-160-20.7-c5-0-8
Degree $2$
Conductor $160$
Sign $0.610 + 0.792i$
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  + (−20.8 − 20.8i)3-s + (−55.6 + 5.74i)5-s + (135. − 135. i)7-s + 624. i·9-s + 629. i·11-s + (−2.08 + 2.08i)13-s + (1.27e3 + 1.03e3i)15-s + (−241. − 241. i)17-s − 372.·19-s − 5.66e3·21-s + (2.03e3 + 2.03e3i)23-s + (3.05e3 − 638. i)25-s + (7.95e3 − 7.95e3i)27-s + 55.2i·29-s − 1.84e3i·31-s + ⋯
L(s)  = 1  + (−1.33 − 1.33i)3-s + (−0.994 + 0.102i)5-s + (1.04 − 1.04i)7-s + 2.57i·9-s + 1.56i·11-s + (−0.00342 + 0.00342i)13-s + (1.46 + 1.19i)15-s + (−0.202 − 0.202i)17-s − 0.236·19-s − 2.80·21-s + (0.801 + 0.801i)23-s + (0.978 − 0.204i)25-s + (2.09 − 2.09i)27-s + 0.0121i·29-s − 0.344i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9179235057\)
\(L(\frac12)\) \(\approx\) \(0.9179235057\)
\(L(\frac{7}{2})\) 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 \( 1 + (55.6 - 5.74i)T \)
good3 \( 1 + (20.8 + 20.8i)T + 243iT^{2} \)
7 \( 1 + (-135. + 135. i)T - 1.68e4iT^{2} \)
11 \( 1 - 629. iT - 1.61e5T^{2} \)
13 \( 1 + (2.08 - 2.08i)T - 3.71e5iT^{2} \)
17 \( 1 + (241. + 241. i)T + 1.41e6iT^{2} \)
19 \( 1 + 372.T + 2.47e6T^{2} \)
23 \( 1 + (-2.03e3 - 2.03e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 55.2iT - 2.05e7T^{2} \)
31 \( 1 + 1.84e3iT - 2.86e7T^{2} \)
37 \( 1 + (61.7 + 61.7i)T + 6.93e7iT^{2} \)
41 \( 1 + 4.80e3T + 1.15e8T^{2} \)
43 \( 1 + (1.06e4 + 1.06e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.21e4 + 1.21e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (2.13e4 - 2.13e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 5.07e4T + 7.14e8T^{2} \)
61 \( 1 - 1.61e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.48e4 + 2.48e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 5.85e4iT - 1.80e9T^{2} \)
73 \( 1 + (4.21e4 - 4.21e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 3.23e4T + 3.07e9T^{2} \)
83 \( 1 + (1.01e4 + 1.01e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 3.22e4iT - 5.58e9T^{2} \)
97 \( 1 + (-2.44e4 - 2.44e4i)T + 8.58e9iT^{2} \)
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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

−11.75766051383527472847028887994, −11.19835183603075062695551851630, −10.29106911127730625953964264568, −8.194970878042195836766210175793, −7.18773603514944974223377016850, −7.04641539769766123805537289489, −5.21640990024723945770514082071, −4.34159804652763624682438158575, −1.84885803388222086020906999088, −0.70417976714304766017299195883, 0.64210427217530464440283275340, 3.34485290380867564546201008395, 4.60190405523343810919785880836, 5.35778472920166291115187288335, 6.42421459021178456300825412056, 8.328810641248786275225613135192, 8.980835364047285723714097607705, 10.50350233844519781618498338104, 11.34133701228146476540452619135, 11.57600806949018470763372910359

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