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 − 4·11-s + 4·13-s − 6·17-s − 4·19-s + 4·23-s + 4·29-s + 4·37-s + 8·41-s + 12·47-s + 9·49-s + 2·53-s + 12·59-s − 2·61-s − 8·67-s − 8·71-s + 16·73-s + 16·77-s + 8·79-s − 8·83-s − 16·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.20·11-s + 1.10·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 0.742·29-s + 0.657·37-s + 1.24·41-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s − 0.949·71-s + 1.87·73-s + 1.82·77-s + 0.900·79-s − 0.878·83-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  =  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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−16.22684173664300, −15.84327071072423, −15.43764029277086, −14.93196339445897, −13.96657118967984, −13.39460589457839, −13.04098512527774, −12.74926564567455, −11.97196997026207, −11.05749431839502, −10.73692893144973, −10.25014230780563, −9.445278422996536, −8.908377843491971, −8.473715446527677, −7.622066475811563, −6.919309262225814, −6.348460075863729, −5.941278153417581, −5.081381302639311, −4.228317013351631, −3.689808678582719, −2.663868383915525, −2.438136360902044, −0.9415531988633392, 0, 0.9415531988633392, 2.438136360902044, 2.663868383915525, 3.689808678582719, 4.228317013351631, 5.081381302639311, 5.941278153417581, 6.348460075863729, 6.919309262225814, 7.622066475811563, 8.473715446527677, 8.908377843491971, 9.445278422996536, 10.25014230780563, 10.73692893144973, 11.05749431839502, 11.97196997026207, 12.74926564567455, 13.04098512527774, 13.39460589457839, 13.96657118967984, 14.93196339445897, 15.43764029277086, 15.84327071072423, 16.22684173664300

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