Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s − 2·7-s + 9-s − 4·11-s − 6·13-s + 2·15-s + 2·17-s + 8·19-s + 4·21-s − 6·23-s + 25-s + 4·27-s − 2·29-s + 4·31-s + 8·33-s + 2·35-s + 2·37-s + 12·39-s − 10·41-s − 2·43-s − 45-s − 2·47-s − 3·49-s − 4·51-s + 2·53-s + 4·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s + 1.83·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s + 1.39·33-s + 0.338·35-s + 0.328·37-s + 1.92·39-s − 1.56·41-s − 0.304·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{160} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 160,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.68105824962705, −18.60280645204874, −17.91787436124905, −16.93129125752339, −16.21822576360131, −15.55547050832315, −14.31489185368755, −13.13143203517445, −12.08463452386232, −11.70642323602708, −10.29364010457156, −9.759510932768573, −7.967338729692306, −7.052696752683617, −5.702434632073582, −4.888475554853501, −3.010643459706773, 0, 3.010643459706773, 4.888475554853501, 5.702434632073582, 7.052696752683617, 7.967338729692306, 9.759510932768573, 10.29364010457156, 11.70642323602708, 12.08463452386232, 13.13143203517445, 14.31489185368755, 15.55547050832315, 16.21822576360131, 16.93129125752339, 17.91787436124905, 18.60280645204874, 19.68105824962705

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