Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.876 + 0.481i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + 0.618·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (−1.30 + 0.951i)11-s + (0.809 + 0.587i)12-s + (0.500 − 0.363i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + 0.618·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (−1.30 + 0.951i)11-s + (0.809 + 0.587i)12-s + (0.500 − 0.363i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s − 0.999·18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.876 + 0.481i$
motivic weight  =  \(0\)
character  :  $\chi_{600} (461, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 600,\ (\ :0),\ 0.876 + 0.481i)$
$L(\frac{1}{2})$  $\approx$  $1.378112086$
$L(\frac12)$  $\approx$  $1.378112086$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
good7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.71161272849659185357909767531, −10.16971236124642587232353105257, −9.437987246934160172361382156219, −8.452096164272570673739495733449, −6.97670638188966200650886100836, −5.69430914124502981247248948137, −5.07288703322210171428009826633, −4.54041723275158527535589520431, −3.10520709311688864591500395430, −1.82616607877136577685655652354, 2.15081326373838919591372847816, 3.03312472714709196783661963924, 4.75093527236746108051610999977, 5.89463443287140223083459360678, 6.02856331054450554983654699819, 7.43258593177356800160634817136, 7.84232125263057793666246361886, 8.863762313301936080766506682848, 10.39695385323574811243124444611, 11.18315070121590300584834859165

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