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 − 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  =  0
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.850684346$
$L(\frac12)$  $\approx$  $1.850684346$
$L(\frac{3}{2})$   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

−16.10839354723984, −15.70722104986870, −14.92405090866785, −14.41677319956909, −13.93940343252848, −13.35715336481839, −12.45717177400896, −12.31730339438544, −11.85605423359099, −10.96739546134274, −10.22417099761109, −9.788693095017872, −9.280172451423393, −8.893422377651092, −7.823261341310853, −7.214247533262595, −6.859109284234373, −5.985344757508338, −5.604132935161850, −4.646074116970887, −3.933998167257511, −3.059036517563198, −2.860558832897748, −1.465288181117594, −0.6320773149724936, 0.6320773149724936, 1.465288181117594, 2.860558832897748, 3.059036517563198, 3.933998167257511, 4.646074116970887, 5.604132935161850, 5.985344757508338, 6.859109284234373, 7.214247533262595, 7.823261341310853, 8.893422377651092, 9.280172451423393, 9.788693095017872, 10.22417099761109, 10.96739546134274, 11.85605423359099, 12.31730339438544, 12.45717177400896, 13.35715336481839, 13.93940343252848, 14.41677319956909, 14.92405090866785, 15.70722104986870, 16.10839354723984

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