Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.911 + 0.410i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.76·3-s + (−43.2 − 35.3i)5-s + 82.4i·7-s − 220.·9-s + 109. i·11-s − 621.·13-s + (206. + 168. i)15-s + 503. i·17-s − 853. i·19-s − 392. i·21-s + 1.42e3i·23-s + (621. + 3.06e3i)25-s + 2.20e3·27-s − 275. i·29-s − 6.20e3·31-s + ⋯
L(s)  = 1  − 0.305·3-s + (−0.774 − 0.632i)5-s + 0.635i·7-s − 0.906·9-s + 0.272i·11-s − 1.02·13-s + (0.236 + 0.193i)15-s + 0.422i·17-s − 0.542i·19-s − 0.194i·21-s + 0.561i·23-s + (0.198 + 0.980i)25-s + 0.583·27-s − 0.0607i·29-s − 1.16·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.911 + 0.410i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.911 + 0.410i)\)
\(L(3)\)  \(\approx\)  \(0.8396202870\)
\(L(\frac12)\)  \(\approx\)  \(0.8396202870\)
\(L(\frac{7}{2})\)   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 \( 1 + (43.2 + 35.3i)T \)
good3 \( 1 + 4.76T + 243T^{2} \)
7 \( 1 - 82.4iT - 1.68e4T^{2} \)
11 \( 1 - 109. iT - 1.61e5T^{2} \)
13 \( 1 + 621.T + 3.71e5T^{2} \)
17 \( 1 - 503. iT - 1.41e6T^{2} \)
19 \( 1 + 853. iT - 2.47e6T^{2} \)
23 \( 1 - 1.42e3iT - 6.43e6T^{2} \)
29 \( 1 + 275. iT - 2.05e7T^{2} \)
31 \( 1 + 6.20e3T + 2.86e7T^{2} \)
37 \( 1 - 3.66e3T + 6.93e7T^{2} \)
41 \( 1 + 1.26e3T + 1.15e8T^{2} \)
43 \( 1 + 7.49e3T + 1.47e8T^{2} \)
47 \( 1 + 1.58e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.00e4T + 4.18e8T^{2} \)
59 \( 1 + 2.79e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.99e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.65e4T + 1.35e9T^{2} \)
71 \( 1 + 3.27e4T + 1.80e9T^{2} \)
73 \( 1 - 7.15e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.07e4T + 3.07e9T^{2} \)
83 \( 1 - 1.08e5T + 3.93e9T^{2} \)
89 \( 1 - 8.48e4T + 5.58e9T^{2} \)
97 \( 1 - 3.83e4iT - 8.58e9T^{2} \)
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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

−10.99308848348307919172296272023, −9.660909643680654506148630163904, −8.812933125064524700382305025324, −7.980495376586717271131470177925, −6.92371099442584064792036525077, −5.56692999600374777491780809890, −4.89211501360526268400790127829, −3.53614485643038582997901520665, −2.16941985088831143415520686217, −0.43269558872863036018848814597, 0.56505450830087583042852612937, 2.54225013321264250947998512396, 3.62654774192449336742969768852, 4.80439500878820810060686718450, 6.02214445108565547143192386723, 7.11454761108384977820186816413, 7.82854174262209079479077588324, 8.927148345739389210889447143121, 10.18373552287826085502893323780, 10.90424466858639012011936365012

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