Properties

Degree 2
Conductor $ 2^{3} \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 + 9-s − 4·11-s − 6·13-s + 6·17-s − 4·19-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 2·37-s + 6·39-s − 6·41-s − 12·43-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s + 4·57-s + 12·59-s + 14·61-s − 4·67-s + 8·71-s + 6·73-s − 8·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.960·39-s − 0.937·41-s − 1.82·43-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{600} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 600,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−10.09854126146590211624669174065, −9.749205121396688870647577836420, −8.280524075884956092273610432842, −7.55727707153378484774141976097, −6.65963768785974737826480192145, −5.36683559095823513937522588289, −4.95639818252911756604562729940, −3.42683606015106566673715003849, −2.07933314479152275364533906758, 0, 2.07933314479152275364533906758, 3.42683606015106566673715003849, 4.95639818252911756604562729940, 5.36683559095823513937522588289, 6.65963768785974737826480192145, 7.55727707153378484774141976097, 8.280524075884956092273610432842, 9.749205121396688870647577836420, 10.09854126146590211624669174065

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