Properties

Degree 2
Conductor $ 2^{4} \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 + 2·13-s + 2·17-s − 4·19-s − 4·23-s + 2·29-s + 8·31-s − 6·37-s + 6·41-s − 8·43-s − 4·47-s + 9·49-s + 6·53-s − 4·59-s − 2·61-s + 8·67-s + 6·73-s − 16·77-s + 16·83-s + 6·89-s − 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s − 0.256·61-s + 0.977·67-s + 0.702·73-s − 1.82·77-s + 1.75·83-s + 0.635·89-s − 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.533094610$
$L(\frac12)$  $\approx$  $1.533094610$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−18.38984877052994, −17.61181977237811, −16.93790270279536, −16.50106583500639, −15.84748205452148, −15.32940496481698, −14.51252457860159, −13.85397944709383, −13.32791770585292, −12.53869203799874, −12.09348942886188, −11.41350999482448, −10.40173317243488, −9.994235907479735, −9.262344366135522, −8.653600111128790, −7.882360368379258, −6.707263499772098, −6.486398942716050, −5.813580465947029, −4.603577247660795, −3.746274879341249, −3.211144560537383, −2.020341685979698, −0.7216988117921999, 0.7216988117921999, 2.020341685979698, 3.211144560537383, 3.746274879341249, 4.603577247660795, 5.813580465947029, 6.486398942716050, 6.707263499772098, 7.882360368379258, 8.653600111128790, 9.262344366135522, 9.994235907479735, 10.40173317243488, 11.41350999482448, 12.09348942886188, 12.53869203799874, 13.32791770585292, 13.85397944709383, 14.51252457860159, 15.32940496481698, 15.84748205452148, 16.50106583500639, 16.93790270279536, 17.61181977237811, 18.38984877052994

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