Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  \(0\)
Selberg data  =  \((2,\ 14400,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.383879945\)
\(L(\frac12)\)  \(\approx\)  \(3.383879945\)
\(L(\frac{3}{2})\)   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

−16.07938445025984, −15.54122210697903, −14.88504265308643, −14.42772704348290, −14.00706282527772, −13.42969658513723, −12.72129623558077, −12.03882013795573, −11.42845617896145, −11.31288528270342, −10.33241558982688, −9.967842305736765, −9.160582353126836, −8.508548825228234, −7.895512663628200, −7.600331878124270, −6.808214055143235, −5.815293094723246, −5.463731643062181, −4.758184075328142, −4.045860436273191, −3.303275308615425, −2.423590410647841, −1.458362549469043, −0.9202905128627810, 0.9202905128627810, 1.458362549469043, 2.423590410647841, 3.303275308615425, 4.045860436273191, 4.758184075328142, 5.463731643062181, 5.815293094723246, 6.808214055143235, 7.600331878124270, 7.895512663628200, 8.508548825228234, 9.160582353126836, 9.967842305736765, 10.33241558982688, 11.31288528270342, 11.42845617896145, 12.03882013795573, 12.72129623558077, 13.42969658513723, 14.00706282527772, 14.42772704348290, 14.88504265308643, 15.54122210697903, 16.07938445025984

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