Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.468 - 0.883i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24.8·3-s + (53.4 − 16.3i)5-s + 100. i·7-s + 372.·9-s − 5.36i·11-s + 506.·13-s + (−1.32e3 + 406. i)15-s + 1.45e3i·17-s + 722. i·19-s − 2.48e3i·21-s − 3.08e3i·23-s + (2.58e3 − 1.75e3i)25-s − 3.20e3·27-s − 7.29e3i·29-s − 8.60e3·31-s + ⋯
L(s)  = 1  − 1.59·3-s + (0.956 − 0.293i)5-s + 0.773i·7-s + 1.53·9-s − 0.0133i·11-s + 0.831·13-s + (−1.52 + 0.466i)15-s + 1.22i·17-s + 0.459i·19-s − 1.23i·21-s − 1.21i·23-s + (0.828 − 0.560i)25-s − 0.846·27-s − 1.61i·29-s − 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.468 - 0.883i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.468 - 0.883i)\)
\(L(3)\)  \(\approx\)  \(1.242889323\)
\(L(\frac12)\)  \(\approx\)  \(1.242889323\)
\(L(\frac{7}{2})\)   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 \( 1 + (-53.4 + 16.3i)T \)
good3 \( 1 + 24.8T + 243T^{2} \)
7 \( 1 - 100. iT - 1.68e4T^{2} \)
11 \( 1 + 5.36iT - 1.61e5T^{2} \)
13 \( 1 - 506.T + 3.71e5T^{2} \)
17 \( 1 - 1.45e3iT - 1.41e6T^{2} \)
19 \( 1 - 722. iT - 2.47e6T^{2} \)
23 \( 1 + 3.08e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.29e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.60e3T + 2.86e7T^{2} \)
37 \( 1 - 5.27e3T + 6.93e7T^{2} \)
41 \( 1 + 1.81e4T + 1.15e8T^{2} \)
43 \( 1 - 8.34e3T + 1.47e8T^{2} \)
47 \( 1 - 2.21e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.33e4T + 4.18e8T^{2} \)
59 \( 1 - 1.09e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.09e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.30e4T + 1.35e9T^{2} \)
71 \( 1 - 3.60e4T + 1.80e9T^{2} \)
73 \( 1 - 6.42e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.86e4T + 3.07e9T^{2} \)
83 \( 1 - 4.97e4T + 3.93e9T^{2} \)
89 \( 1 - 7.76e3T + 5.58e9T^{2} \)
97 \( 1 + 3.68e4iT - 8.58e9T^{2} \)
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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

−10.88756519341313571714926765022, −10.30811438015142731601231697998, −9.225167568251774248573374980059, −8.202725493391364812545195446481, −6.53850073488637498941981470046, −5.97653582539242631745224277828, −5.36556552023723522252217208524, −4.13072401236456434608762002470, −2.14639742736100374746648384997, −0.952449072385739289890792611965, 0.51842897044945029400819132120, 1.61890108076752489292556942430, 3.49757583319129793911172084449, 5.02454798062844844802051574917, 5.59050007580225073208724834204, 6.72182273992891342827832080232, 7.23926778295929276641112195919, 9.039106434370135146921681606891, 9.972716461213919278739053566635, 10.86160770125124525400206984796

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