Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
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  + 15.6i·3-s − 25i·5-s + 164.·7-s − 2.80·9-s + 744. i·11-s − 873. i·13-s + 391.·15-s + 984.·17-s − 1.14e3i·19-s + 2.58e3i·21-s − 578.·23-s − 625·25-s + 3.76e3i·27-s − 5.86e3i·29-s + 9.82e3·31-s + ⋯
L(s)  = 1  + 1.00i·3-s − 0.447i·5-s + 1.27·7-s − 0.0115·9-s + 1.85i·11-s − 1.43i·13-s + 0.449·15-s + 0.826·17-s − 0.730i·19-s + 1.27i·21-s − 0.227·23-s − 0.200·25-s + 0.994i·27-s − 1.29i·29-s + 1.83·31-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

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.707 - 0.707i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (161, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.707 - 0.707i)\)
\(L(3)\)  \(\approx\)  \(2.721855017\)
\(L(\frac12)\)  \(\approx\)  \(2.721855017\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 25iT \)
good3 \( 1 - 15.6iT - 243T^{2} \)
7 \( 1 - 164.T + 1.68e4T^{2} \)
11 \( 1 - 744. iT - 1.61e5T^{2} \)
13 \( 1 + 873. iT - 3.71e5T^{2} \)
17 \( 1 - 984.T + 1.41e6T^{2} \)
19 \( 1 + 1.14e3iT - 2.47e6T^{2} \)
23 \( 1 + 578.T + 6.43e6T^{2} \)
29 \( 1 + 5.86e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.82e3T + 2.86e7T^{2} \)
37 \( 1 + 7.88e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
43 \( 1 + 1.90e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.97e3T + 2.29e8T^{2} \)
53 \( 1 - 3.59e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.93e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.78e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.13e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.39e4T + 1.80e9T^{2} \)
73 \( 1 - 2.27e4T + 2.07e9T^{2} \)
79 \( 1 - 9.47e4T + 3.07e9T^{2} \)
83 \( 1 - 3.47e4iT - 3.93e9T^{2} \)
89 \( 1 + 4.20e4T + 5.58e9T^{2} \)
97 \( 1 + 1.14e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.61506688451263369844454856873, −10.09409261217699730866932158108, −9.222241017545083433797503271051, −8.025643441548117015661390950244, −7.37250621420961928911998268421, −5.58762313664993604560873010818, −4.74269900544334624742346381060, −4.15523544760009424551426541576, −2.42906816839741649209519472676, −1.00960108158720347511503848113, 0.973604708196961233250275781199, 1.80994834319758414303867646790, 3.26902340153684183744134148897, 4.67505607492585877485508286499, 6.02308081050548451448827771380, 6.76567173137055249514619784855, 8.000541388965954919859125998868, 8.335513772826080864234450113593, 9.771375418254780730695960430544, 11.03163412914139744304179728596

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