Properties

Degree 2
Conductor $ 2^{6} \cdot 5^{2} $
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 + 2·7-s + 9-s + 2·13-s + 6·17-s − 4·19-s + 4·21-s + 6·23-s − 4·27-s − 6·29-s + 4·31-s + 2·37-s + 4·39-s + 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s + 2·63-s − 2·67-s + 12·69-s + 12·71-s − 2·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-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 + 0.251·63-s − 0.244·67-s + 1.44·69-s + 1.42·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1600\)    =    \(2^{6} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.876616721$
$L(\frac12)$  $\approx$  $2.876616721$
$L(\frac{3}{2})$   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 \( 1 \)
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

−19.82179894603704, −19.07677394337119, −18.88270911541747, −17.90294948990113, −17.19116666231699, −16.58685005158953, −15.66300635991939, −14.93986419610657, −14.46786656501638, −14.04089471793946, −13.07253477298963, −12.63567475470350, −11.44806299752069, −11.02497315560520, −9.997756160900699, −9.249279365224539, −8.550149092441661, −7.950176007435402, −7.332796216477512, −6.141811606848554, −5.259742312761404, −4.190978474931607, −3.328879588478321, −2.416628817188111, −1.265750594913709, 1.265750594913709, 2.416628817188111, 3.328879588478321, 4.190978474931607, 5.259742312761404, 6.141811606848554, 7.332796216477512, 7.950176007435402, 8.550149092441661, 9.249279365224539, 9.997756160900699, 11.02497315560520, 11.44806299752069, 12.63567475470350, 13.07253477298963, 14.04089471793946, 14.46786656501638, 14.93986419610657, 15.66300635991939, 16.58685005158953, 17.19116666231699, 17.90294948990113, 18.88270911541747, 19.07677394337119, 19.82179894603704

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