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  + 21.6i·3-s − 25i·5-s − 169.·7-s − 224.·9-s − 512. i·11-s − 36.4i·13-s + 540.·15-s − 1.26e3·17-s + 790. i·19-s − 3.66e3i·21-s + 4.99e3·23-s − 625·25-s + 401. i·27-s + 934. i·29-s + 6.69e3·31-s + ⋯
L(s)  = 1  + 1.38i·3-s − 0.447i·5-s − 1.30·7-s − 0.923·9-s − 1.27i·11-s − 0.0598i·13-s + 0.620·15-s − 1.05·17-s + 0.502i·19-s − 1.81i·21-s + 1.96·23-s − 0.200·25-s + 0.106i·27-s + 0.206i·29-s + 1.25·31-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

\(L(3)\) \(\approx\) \(1.483656199\)
\(L(\frac12)\) \(\approx\) \(1.483656199\)
\(L(\frac{7}{2})\) 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 \( 1 - 21.6iT - 243T^{2} \)
7 \( 1 + 169.T + 1.68e4T^{2} \)
11 \( 1 + 512. iT - 1.61e5T^{2} \)
13 \( 1 + 36.4iT - 3.71e5T^{2} \)
17 \( 1 + 1.26e3T + 1.41e6T^{2} \)
19 \( 1 - 790. iT - 2.47e6T^{2} \)
23 \( 1 - 4.99e3T + 6.43e6T^{2} \)
29 \( 1 - 934. iT - 2.05e7T^{2} \)
31 \( 1 - 6.69e3T + 2.86e7T^{2} \)
37 \( 1 + 4.30e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.03e4T + 1.15e8T^{2} \)
43 \( 1 - 6.37e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.21e4T + 2.29e8T^{2} \)
53 \( 1 + 2.32e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.77e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.99e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.54e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.98e4T + 1.80e9T^{2} \)
73 \( 1 + 435.T + 2.07e9T^{2} \)
79 \( 1 + 6.16e4T + 3.07e9T^{2} \)
83 \( 1 - 2.94e3iT - 3.93e9T^{2} \)
89 \( 1 - 6.47e4T + 5.58e9T^{2} \)
97 \( 1 - 1.59e5T + 8.58e9T^{2} \)
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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

−10.73815998735997763507907612163, −9.968588823736185335237431841013, −9.082527076286042027554335479488, −8.576974510600062909538269585384, −6.87497974501507718872713462218, −5.83929869812138479403060197499, −4.81440405908055969236320330557, −3.71738392175713744946768507023, −2.91860886808259766676361948715, −0.65095184621919858008266200682, 0.67255185256065740448129217414, 2.11761132718205947878400243489, 3.03612911067294077104910799486, 4.66000573876479079074735998652, 6.27873656536553628737719409244, 6.86827289382220757817772447900, 7.38782648165554503754171025618, 8.780942419417228612815915490930, 9.698016547477090520834433638836, 10.72513223677904618634824139695

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