Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.991 - 0.130i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.965 − 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (−0.258 − 0.965i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (−0.258 + 0.965i)18-s + i·19-s + (1.22 + 1.22i)22-s + (0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.965 − 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (−0.258 − 0.965i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (−0.258 + 0.965i)18-s + i·19-s + (1.22 + 1.22i)22-s + (0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.991 - 0.130i$
motivic weight  =  \(0\)
character  :  $\chi_{600} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :0),\ 0.991 - 0.130i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.8858299358\)
\(L(\frac12)\)  \(\approx\)  \(0.8858299358\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.65427956932928628926149291078, −9.786658380750722333654077956878, −8.834226499925666687571611685444, −8.340207736389128160026029759172, −7.62970450650671747037675368682, −6.46893148745807972454904486372, −5.82769030661011163315792003108, −4.29461748777089127943875183278, −3.00787396981305638761693367608, −1.47224940725126572218738898426, 1.91540564081001050958220064208, 2.76206245112357439677472686171, 4.08650844877286557769597642430, 4.84021912317408969478257054777, 7.05094998844366798044334666795, 7.31868632952487026059281293649, 8.553362702842214825505602013615, 9.193505144241107666651545853368, 9.882752347528073464154965589982, 10.58466898194519248950448277548

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