Properties

Degree $2$
Conductor $320$
Sign $0.707 + 0.707i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.72i·3-s + 25i·5-s + 65.1·7-s + 148.·9-s − 71.5i·11-s − 733. i·13-s − 243.·15-s − 716.·17-s − 2.33e3i·19-s + 633. i·21-s − 278.·23-s − 625·25-s + 3.80e3i·27-s − 2.95e3i·29-s − 6.04e3·31-s + ⋯
L(s)  = 1  + 0.623i·3-s + 0.447i·5-s + 0.502·7-s + 0.610·9-s − 0.178i·11-s − 1.20i·13-s − 0.279·15-s − 0.600·17-s − 1.48i·19-s + 0.313i·21-s − 0.109·23-s − 0.200·25-s + 1.00i·27-s − 0.652i·29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.707 + 0.707i$
Motivic weight: \(5\)
Character: $\chi_{320} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.842657892\)
\(L(\frac12)\) \(\approx\) \(1.842657892\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.72iT - 243T^{2} \)
7 \( 1 - 65.1T + 1.68e4T^{2} \)
11 \( 1 + 71.5iT - 1.61e5T^{2} \)
13 \( 1 + 733. iT - 3.71e5T^{2} \)
17 \( 1 + 716.T + 1.41e6T^{2} \)
19 \( 1 + 2.33e3iT - 2.47e6T^{2} \)
23 \( 1 + 278.T + 6.43e6T^{2} \)
29 \( 1 + 2.95e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.04e3T + 2.86e7T^{2} \)
37 \( 1 + 1.21e4iT - 6.93e7T^{2} \)
41 \( 1 - 3.62e3T + 1.15e8T^{2} \)
43 \( 1 - 230. iT - 1.47e8T^{2} \)
47 \( 1 - 1.39e4T + 2.29e8T^{2} \)
53 \( 1 + 2.24e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.28e3iT - 7.14e8T^{2} \)
61 \( 1 + 1.94e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.39e3iT - 1.35e9T^{2} \)
71 \( 1 - 5.73e4T + 1.80e9T^{2} \)
73 \( 1 + 3.55e4T + 2.07e9T^{2} \)
79 \( 1 - 9.30e4T + 3.07e9T^{2} \)
83 \( 1 - 1.59e3iT - 3.93e9T^{2} \)
89 \( 1 - 2.02e3T + 5.58e9T^{2} \)
97 \( 1 + 1.20e5T + 8.58e9T^{2} \)
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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

−10.83671720511157391392977170516, −9.810011480661996627104578116276, −8.934980567824998230980315243593, −7.77659650290398804869257173251, −6.90345387716106616167022264325, −5.58034533650174983424526081802, −4.60598454269606263978082954339, −3.51251526353820729014078489872, −2.20970181028919976607459147024, −0.50275881414376860028670259820, 1.27768887615432715781919701538, 2.01514737070353161492225991134, 3.90316657463457265188861634287, 4.83286297541393707061905561011, 6.15650044497481256161781527330, 7.12023362328182720119354219812, 7.989698485705439041842817682477, 8.971247731666202202748333984661, 9.924606699954945606648123792728, 11.02172839761618987488366793860

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