Properties

Degree $2$
Conductor $320$
Sign $-0.965 + 0.258i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.5i·3-s + 25i·5-s − 55.4·7-s − 31.1·9-s + 153. i·11-s + 57.3i·13-s + 413.·15-s + 1.78e3·17-s − 1.59e3i·19-s + 918. i·21-s − 4.11e3·23-s − 625·25-s − 3.50e3i·27-s − 5.00e3i·29-s − 1.82e3·31-s + ⋯
L(s)  = 1  − 1.06i·3-s + 0.447i·5-s − 0.427·7-s − 0.128·9-s + 0.382i·11-s + 0.0940i·13-s + 0.475·15-s + 1.49·17-s − 1.01i·19-s + 0.454i·21-s − 1.62·23-s − 0.200·25-s − 0.925i·27-s − 1.10i·29-s − 0.340·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.965 + 0.258i$
Motivic weight: \(5\)
Character: $\chi_{320} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9993862735\)
\(L(\frac12)\) \(\approx\) \(0.9993862735\)
\(L(\frac{7}{2})\) 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 \( 1 - 25iT \)
good3 \( 1 + 16.5iT - 243T^{2} \)
7 \( 1 + 55.4T + 1.68e4T^{2} \)
11 \( 1 - 153. iT - 1.61e5T^{2} \)
13 \( 1 - 57.3iT - 3.71e5T^{2} \)
17 \( 1 - 1.78e3T + 1.41e6T^{2} \)
19 \( 1 + 1.59e3iT - 2.47e6T^{2} \)
23 \( 1 + 4.11e3T + 6.43e6T^{2} \)
29 \( 1 + 5.00e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.82e3T + 2.86e7T^{2} \)
37 \( 1 - 1.80e3iT - 6.93e7T^{2} \)
41 \( 1 - 5.45e3T + 1.15e8T^{2} \)
43 \( 1 + 8.25e3iT - 1.47e8T^{2} \)
47 \( 1 + 8.40e3T + 2.29e8T^{2} \)
53 \( 1 + 6.74e3iT - 4.18e8T^{2} \)
59 \( 1 + 17.5iT - 7.14e8T^{2} \)
61 \( 1 - 1.88e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.97e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.68e4T + 1.80e9T^{2} \)
73 \( 1 + 5.66e4T + 2.07e9T^{2} \)
79 \( 1 + 3.72e4T + 3.07e9T^{2} \)
83 \( 1 + 5.78e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.31e4T + 5.58e9T^{2} \)
97 \( 1 + 9.12e4T + 8.58e9T^{2} \)
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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

−10.23930529873486129525501043687, −9.604467405970492152734004193658, −8.156505864127968206193517601330, −7.42842025337638863372704071571, −6.60097563407526832286117773503, −5.69387792845239613347957017798, −4.12184225586284327058626147355, −2.77046799936795646809726219999, −1.62856870754518905643971614165, −0.26395195007646110864636698218, 1.40603603756712659564852137878, 3.26554012045303711294523483816, 4.04592064682862567714265384348, 5.24197053279454247530523240591, 6.07751607676997700517780841570, 7.58572070286525733314143608078, 8.510176055358193198153043337906, 9.686154148378528157012371475686, 10.02196082969068156534199719976, 11.03865797616050680675212079218

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