Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \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  − 5·7-s − 6·11-s + 3·13-s + 2·17-s − 19-s + 2·23-s + 6·29-s + 3·31-s + 6·37-s − 4·41-s − 11·43-s + 10·47-s + 18·49-s − 8·53-s − 6·59-s − 3·61-s + 67-s + 12·71-s + 10·73-s + 30·77-s − 8·79-s − 6·83-s + 16·89-s − 15·91-s − 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.88·7-s − 1.80·11-s + 0.832·13-s + 0.485·17-s − 0.229·19-s + 0.417·23-s + 1.11·29-s + 0.538·31-s + 0.986·37-s − 0.624·41-s − 1.67·43-s + 1.45·47-s + 18/7·49-s − 1.09·53-s − 0.781·59-s − 0.384·61-s + 0.122·67-s + 1.42·71-s + 1.17·73-s + 3.41·77-s − 0.900·79-s − 0.658·83-s + 1.69·89-s − 1.57·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{14400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−16.19546664262736, −15.87968543369966, −15.46369349716959, −14.94376464305954, −13.82587222467119, −13.56614497766053, −13.06765200445376, −12.52575203324912, −12.14372406899814, −11.14331580324988, −10.60590285366821, −10.08676840016979, −9.694724980154990, −8.946636699385996, −8.248296320697156, −7.764148138097945, −6.871633070997952, −6.429112882515141, −5.807462361919975, −5.162119118117353, −4.337974131316629, −3.340996918180631, −3.039455562142013, −2.301749682128527, −0.9165565051626417, 0, 0.9165565051626417, 2.301749682128527, 3.039455562142013, 3.340996918180631, 4.337974131316629, 5.162119118117353, 5.807462361919975, 6.429112882515141, 6.871633070997952, 7.764148138097945, 8.248296320697156, 8.946636699385996, 9.694724980154990, 10.08676840016979, 10.60590285366821, 11.14331580324988, 12.14372406899814, 12.52575203324912, 13.06765200445376, 13.56614497766053, 13.82587222467119, 14.94376464305954, 15.46369349716959, 15.87968543369966, 16.19546664262736

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