Properties

Degree 2
Conductor $ 2^{4} \cdot 5 $
Sign $-0.406 - 0.913i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 179. i·3-s + (−568. − 1.27e3i)5-s − 8.71e3i·7-s − 1.24e4·9-s − 4.45e4·11-s + 2.14e4i·13-s + (2.28e5 − 1.01e5i)15-s + 3.00e5i·17-s + 5.65e5·19-s + 1.56e6·21-s + 9.50e5i·23-s + (−1.30e6 + 1.45e6i)25-s + 1.29e6i·27-s + 8.03e5·29-s + 1.99e6·31-s + ⋯
L(s)  = 1  + 1.27i·3-s + (−0.406 − 0.913i)5-s − 1.37i·7-s − 0.632·9-s − 0.917·11-s + 0.208i·13-s + (1.16 − 0.519i)15-s + 0.871i·17-s + 0.995·19-s + 1.75·21-s + 0.708i·23-s + (−0.669 + 0.742i)25-s + 0.469i·27-s + 0.210·29-s + 0.388·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $-0.406 - 0.913i$
motivic weight  =  \(9\)
character  :  $\chi_{80} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 80,\ (\ :9/2),\ -0.406 - 0.913i)$
$L(5)$  $\approx$  $0.617387 + 0.950576i$
$L(\frac12)$  $\approx$  $0.617387 + 0.950576i$
$L(\frac{11}{2})$   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 \( 1 + (568. + 1.27e3i)T \)
good3 \( 1 - 179. iT - 1.96e4T^{2} \)
7 \( 1 + 8.71e3iT - 4.03e7T^{2} \)
11 \( 1 + 4.45e4T + 2.35e9T^{2} \)
13 \( 1 - 2.14e4iT - 1.06e10T^{2} \)
17 \( 1 - 3.00e5iT - 1.18e11T^{2} \)
19 \( 1 - 5.65e5T + 3.22e11T^{2} \)
23 \( 1 - 9.50e5iT - 1.80e12T^{2} \)
29 \( 1 - 8.03e5T + 1.45e13T^{2} \)
31 \( 1 - 1.99e6T + 2.64e13T^{2} \)
37 \( 1 - 9.53e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.54e7T + 3.27e14T^{2} \)
43 \( 1 + 2.32e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.77e7iT - 1.11e15T^{2} \)
53 \( 1 - 4.79e7iT - 3.29e15T^{2} \)
59 \( 1 - 7.00e7T + 8.66e15T^{2} \)
61 \( 1 - 1.26e8T + 1.16e16T^{2} \)
67 \( 1 - 2.66e8iT - 2.72e16T^{2} \)
71 \( 1 + 6.59e7T + 4.58e16T^{2} \)
73 \( 1 + 1.47e7iT - 5.88e16T^{2} \)
79 \( 1 + 4.66e7T + 1.19e17T^{2} \)
83 \( 1 - 2.01e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.54e8T + 3.50e17T^{2} \)
97 \( 1 - 3.39e8iT - 7.60e17T^{2} \)
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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

−12.98773176174370843404940343057, −11.56778355904474761781302216769, −10.44453957041421670909206855803, −9.785167312625617822178874126682, −8.462707599255700230685914409530, −7.33394130276484547415730213514, −5.33897631880823323120380488811, −4.38260877871394258121937534592, −3.50421662057204139631572919415, −1.12537785955381932095631722663, 0.35934690874260760425372986619, 2.15830639741145413924932622528, 2.98790541399688131350941171475, 5.28945617470984078496631646545, 6.51736075083567173522745182658, 7.50416167251085573976948489270, 8.442228961069140197960821438901, 10.00851752410342313310950617424, 11.46960829190914062033662997549, 12.14068911930395952696424218646

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