Properties

Degree 2
Conductor $ 2^{6} \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 − 2·13-s + 2·15-s − 6·17-s + 4·19-s + 4·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s − 4·31-s + 2·35-s − 2·37-s − 4·39-s + 6·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s − 12·51-s + 6·53-s + 8·57-s − 12·59-s − 2·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{320} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 320,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.997134815$
$L(\frac12)$  $\approx$  $1.997134815$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.93857312743352, −19.33484027710300, −18.31110639172838, −17.62448488727226, −16.83476081709403, −15.65443399595050, −14.86032005438100, −14.32311277628388, −13.50772782414131, −12.81052870509687, −11.48400710378320, −10.75706369377723, −9.332287203335425, −9.039858624083401, −7.874894595742304, −7.094649403601694, −5.594815389252715, −4.451949017722053, −3.014366117506322, −1.912026946143006, 1.912026946143006, 3.014366117506322, 4.451949017722053, 5.594815389252715, 7.094649403601694, 7.874894595742304, 9.039858624083401, 9.332287203335425, 10.75706369377723, 11.48400710378320, 12.81052870509687, 13.50772782414131, 14.32311277628388, 14.86032005438100, 15.65443399595050, 16.83476081709403, 17.62448488727226, 18.31110639172838, 19.33484027710300, 19.93857312743352

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