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 − 4·11-s + 6·13-s + 2·17-s + 4·19-s + 10·29-s + 4·31-s − 10·37-s − 2·41-s + 4·43-s − 8·47-s + 9·49-s − 2·53-s − 12·59-s + 10·61-s − 12·67-s − 10·73-s + 16·77-s + 4·79-s + 4·83-s + 6·89-s − 24·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.85·29-s + 0.718·31-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s − 1.56·59-s + 1.28·61-s − 1.46·67-s − 1.17·73-s + 1.82·77-s + 0.450·79-s + 0.439·83-s + 0.635·89-s − 2.51·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.529208041$
$L(\frac12)$  $\approx$  $1.529208041$
$L(\frac{3}{2})$   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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.16772721683583, −15.71325119683406, −15.31277365511338, −14.22716834789009, −13.77821880996657, −13.29298460315862, −12.90268332912593, −12.16029168995681, −11.75719886283108, −10.82043546518578, −10.30997131799555, −10.02652029851156, −9.190123267784658, −8.643764418259445, −8.022671535721797, −7.365570059099450, −6.487142720872989, −6.249220151101373, −5.454422971820345, −4.803725686605519, −3.790997775715735, −3.139169321814783, −2.832866758616042, −1.505772572463631, −0.5597348109658727, 0.5597348109658727, 1.505772572463631, 2.832866758616042, 3.139169321814783, 3.790997775715735, 4.803725686605519, 5.454422971820345, 6.249220151101373, 6.487142720872989, 7.365570059099450, 8.022671535721797, 8.643764418259445, 9.190123267784658, 10.02652029851156, 10.30997131799555, 10.82043546518578, 11.75719886283108, 12.16029168995681, 12.90268332912593, 13.29298460315862, 13.77821880996657, 14.22716834789009, 15.31277365511338, 15.71325119683406, 16.16772721683583

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