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 + 4·13-s − 8·19-s − 4·23-s − 6·29-s + 8·31-s − 4·37-s − 6·41-s − 4·43-s + 4·47-s + 9·49-s + 12·53-s + 6·61-s − 12·67-s − 16·71-s + 8·79-s − 12·83-s + 10·89-s − 16·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.10·13-s − 1.83·19-s − 0.834·23-s − 1.11·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 1.64·53-s + 0.768·61-s − 1.46·67-s − 1.89·71-s + 0.900·79-s − 1.31·83-s + 1.05·89-s − 1.67·91-s − 0.812·97-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  =  0
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9848188363$
$L(\frac12)$  $\approx$  $0.9848188363$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.10773647107597, −15.56652219717795, −15.15503803541259, −14.49868460498522, −13.59629930141832, −13.32487686416598, −12.92910842482682, −12.12746122447521, −11.76903523491186, −10.86080866184508, −10.29841460837396, −10.01993490143958, −9.117680692737109, −8.668585453129924, −8.162250928545318, −7.166474518035704, −6.650292950005909, −6.075683775011601, −5.699363979127569, −4.532586147431031, −3.904970932396257, −3.366056335891529, −2.522573561735800, −1.677918090442625, −0.4230746157945803, 0.4230746157945803, 1.677918090442625, 2.522573561735800, 3.366056335891529, 3.904970932396257, 4.532586147431031, 5.699363979127569, 6.075683775011601, 6.650292950005909, 7.166474518035704, 8.162250928545318, 8.668585453129924, 9.117680692737109, 10.01993490143958, 10.29841460837396, 10.86080866184508, 11.76903523491186, 12.12746122447521, 12.92910842482682, 13.32487686416598, 13.59629930141832, 14.49868460498522, 15.15503803541259, 15.56652219717795, 16.10773647107597

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