Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  0
Selberg data  =  $(2,\ 160,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.505776227$
$L(\frac12)$  $\approx$  $1.505776227$
$L(\frac{3}{2})$   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

−19.86613337683485, −19.45899210944543, −18.69333252856616, −17.08635501954577, −16.98281114909974, −15.10649913002272, −14.80319689853036, −14.20093172716624, −12.90166435982223, −11.96543784493566, −10.94344180239528, −9.572587619444812, −8.731345173958243, −7.851407547172465, −6.819512693219067, −4.929455252585614, −3.695186109831281, −2.173418171649044, 2.173418171649044, 3.695186109831281, 4.929455252585614, 6.819512693219067, 7.851407547172465, 8.731345173958243, 9.572587619444812, 10.94344180239528, 11.96543784493566, 12.90166435982223, 14.20093172716624, 14.80319689853036, 15.10649913002272, 16.98281114909974, 17.08635501954577, 18.69333252856616, 19.45899210944543, 19.86613337683485

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