Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.717 - 0.696i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.144 + 1.40i)2-s − 3-s + (−1.95 − 0.406i)4-s + (0.144 − 1.40i)6-s − 3.62i·7-s + (0.855 − 2.69i)8-s + 9-s + 6.20i·11-s + (1.95 + 0.406i)12-s − 0.578·13-s + (5.10 + 0.524i)14-s + (3.66 + 1.59i)16-s + 1.42i·17-s + (−0.144 + 1.40i)18-s + 5.62i·19-s + ⋯
L(s)  = 1  + (−0.102 + 0.994i)2-s − 0.577·3-s + (−0.979 − 0.203i)4-s + (0.0590 − 0.574i)6-s − 1.37i·7-s + (0.302 − 0.953i)8-s + 0.333·9-s + 1.87i·11-s + (0.565 + 0.117i)12-s − 0.160·13-s + (1.36 + 0.140i)14-s + (0.917 + 0.398i)16-s + 0.344i·17-s + (−0.0340 + 0.331i)18-s + 1.29i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.717 - 0.696i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ -0.717 - 0.696i)\)
\(L(1)\)  \(\approx\)  \(0.300045 + 0.739497i\)
\(L(\frac12)\)  \(\approx\)  \(0.300045 + 0.739497i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) 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 \( 1 + (0.144 - 1.40i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 3.62iT - 7T^{2} \)
11 \( 1 - 6.20iT - 11T^{2} \)
13 \( 1 + 0.578T + 13T^{2} \)
17 \( 1 - 1.42iT - 17T^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 - 6.78iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 8.41T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 5.42T + 79T^{2} \)
83 \( 1 - 3.25T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 4.84iT - 97T^{2} \)
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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

−10.68478174294418379747816060220, −9.967124015259165221254840775192, −9.473566897626017550280921785378, −7.85476970944891725221182309500, −7.45922572358450266777781150226, −6.64609790711730922408455178439, −5.62127773430141346814980798220, −4.51384057681432099411117707924, −3.93153066843160318929114544702, −1.41435310844905832860546651901, 0.54388433427110706472065004062, 2.37782268559277801117909772703, 3.29705382881241156709347973515, 4.77537189873276606493055123896, 5.55728717117692810265656600414, 6.46401306264438788070095442518, 8.070886150020945909963537640349, 8.836255929669702365732464080210, 9.409860630105957912173617213996, 10.64675645213611373785847268377

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