Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7 $
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 + 7-s + 9-s + 4·13-s − 6·17-s − 2·19-s − 2·21-s + 4·27-s − 6·29-s + 4·31-s − 2·37-s − 8·39-s + 6·41-s + 8·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s + 4·57-s + 6·59-s + 8·61-s + 63-s − 4·67-s − 2·73-s − 8·79-s − 11·81-s − 6·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 0.529·57-s + 0.781·59-s + 1.02·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s − 0.900·79-s − 1.22·81-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2800} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2800,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.533019216248269268759734104285, −7.60707532453471338039224955289, −6.60348506563741211247253098956, −6.19185902002877876721827038865, −5.38990980084792703269563412867, −4.58557293909771964767836479942, −3.83664653997120963252560991499, −2.50419401332512614244886579026, −1.31920416816215806650160602271, 0, 1.31920416816215806650160602271, 2.50419401332512614244886579026, 3.83664653997120963252560991499, 4.58557293909771964767836479942, 5.38990980084792703269563412867, 6.19185902002877876721827038865, 6.60348506563741211247253098956, 7.60707532453471338039224955289, 8.533019216248269268759734104285

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