Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \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  + 3-s − 4·7-s + 9-s − 2·13-s − 6·17-s + 4·19-s − 4·21-s + 27-s − 6·29-s − 8·31-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 9·49-s − 6·51-s + 6·53-s + 4·57-s − 10·61-s − 4·63-s − 4·67-s − 2·73-s − 8·79-s + 81-s + 12·83-s − 6·87-s + 18·89-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s − 0.488·67-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1200,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.65116496156121, −18.78186332793687, −18.30340754211936, −17.37313752272590, −16.52725372945612, −16.04253340297848, −15.27757880127058, −14.73279773921233, −13.61556401637479, −13.29474489175565, −12.57890216246174, −11.76663322814362, −10.77813887953294, −9.957952771932102, −9.276841907117527, −8.811879039450230, −7.530234817144250, −6.971212592332550, −6.108977733147342, −5.031862352115197, −3.828396350627021, −3.107142368048668, −2.014623550445296, 0, 2.014623550445296, 3.107142368048668, 3.828396350627021, 5.031862352115197, 6.108977733147342, 6.971212592332550, 7.530234817144250, 8.811879039450230, 9.276841907117527, 9.957952771932102, 10.77813887953294, 11.76663322814362, 12.57890216246174, 13.29474489175565, 13.61556401637479, 14.73279773921233, 15.27757880127058, 16.04253340297848, 16.52725372945612, 17.37313752272590, 18.30340754211936, 18.78186332793687, 19.65116496156121

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