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  − 4·7-s − 2·13-s − 6·17-s + 4·23-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s − 12·43-s + 12·47-s + 9·49-s + 10·53-s − 8·59-s + 10·61-s + 12·67-s + 8·71-s − 10·73-s − 16·79-s + 12·83-s + 6·89-s + 8·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s − 1.45·17-s + 0.834·23-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 1.82·43-s + 1.75·47-s + 9/7·49-s + 1.37·53-s − 1.04·59-s + 1.28·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 1.31·83-s + 0.635·89-s + 0.838·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.34426519724175, −15.82178258744746, −15.30278598078194, −14.92918773866209, −14.03533874839740, −13.47038544346759, −13.02830013438351, −12.64091384262430, −11.87613388942812, −11.36798005791316, −10.62287504215641, −10.06444296623140, −9.532403001544585, −9.006073167477912, −8.437971516614493, −7.549533373051492, −6.867068420513531, −6.536541258298590, −5.850994524038220, −5.039851678986195, −4.291167996189038, −3.658211869926295, −2.707406983675103, −2.390884703420168, −0.9550165607083523, 0, 0.9550165607083523, 2.390884703420168, 2.707406983675103, 3.658211869926295, 4.291167996189038, 5.039851678986195, 5.850994524038220, 6.536541258298590, 6.867068420513531, 7.549533373051492, 8.437971516614493, 9.006073167477912, 9.532403001544585, 10.06444296623140, 10.62287504215641, 11.36798005791316, 11.87613388942812, 12.64091384262430, 13.02830013438351, 13.47038544346759, 14.03533874839740, 14.92918773866209, 15.30278598078194, 15.82178258744746, 16.34426519724175

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