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 − 4·19-s − 6·29-s − 8·31-s + 2·37-s + 6·41-s + 4·43-s + 9·49-s + 6·53-s + 10·61-s + 4·67-s − 2·73-s − 8·79-s + 12·83-s − 18·89-s − 8·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 24·119-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.824·53-s + 1.28·61-s + 0.488·67-s − 0.234·73-s − 0.900·79-s + 1.31·83-s − 1.90·89-s − 0.838·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 2.20·119-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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−16.28839810493300, −16.05065129345175, −15.30839682467418, −14.64453712880203, −14.31042217946360, −13.36969759702698, −13.03352396612403, −12.56651447374081, −12.05875053465977, −11.16846312452313, −10.75427426446881, −9.981795991377771, −9.604292489562196, −9.007289278990736, −8.377174523222303, −7.526794907693353, −7.097634452433220, −6.281271162832570, −5.821266103107618, −5.279388333746701, −4.024825326278763, −3.723994701877763, −2.966575692451078, −2.147399758773100, −1.033610003952489, 0, 1.033610003952489, 2.147399758773100, 2.966575692451078, 3.723994701877763, 4.024825326278763, 5.279388333746701, 5.821266103107618, 6.281271162832570, 7.097634452433220, 7.526794907693353, 8.377174523222303, 9.007289278990736, 9.604292489562196, 9.981795991377771, 10.75427426446881, 11.16846312452313, 12.05875053465977, 12.56651447374081, 13.03352396612403, 13.36969759702698, 14.31042217946360, 14.64453712880203, 15.30839682467418, 16.05065129345175, 16.28839810493300

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