Properties

Degree 2
Conductor $ 2^{6} \cdot 5^{2} $
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 − 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  =  1
Selberg data  =  $(2,\ 1600,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ 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

−19.47675744045876, −18.58607436520331, −18.28307016384592, −17.50909957133753, −16.68381969845360, −16.36479428621821, −15.84026248815131, −14.84167097210988, −14.08703878451731, −13.35239103215597, −12.47534446500231, −12.04659429347285, −11.30807115484156, −10.62128336551183, −9.841515918649496, −9.258669147089765, −8.059368180384025, −7.355395719905968, −6.297012584279271, −5.824008243593601, −5.135877446683605, −3.887297486586859, −3.052439896826099, −1.387985380481637, 0, 1.387985380481637, 3.052439896826099, 3.887297486586859, 5.135877446683605, 5.824008243593601, 6.297012584279271, 7.355395719905968, 8.059368180384025, 9.258669147089765, 9.841515918649496, 10.62128336551183, 11.30807115484156, 12.04659429347285, 12.47534446500231, 13.35239103215597, 14.08703878451731, 14.84167097210988, 15.84026248815131, 16.36479428621821, 16.68381969845360, 17.50909957133753, 18.28307016384592, 18.58607436520331, 19.47675744045876

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