Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} $
Sign $0.976 - 0.216i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−1.22 − 1.22i)3-s + (−0.866 + 0.499i)4-s + (−0.866 + 1.49i)6-s + (−4.57 + 1.22i)7-s + (0.707 + 0.707i)8-s + 2.99i·9-s + (3 + 1.73i)11-s + (1.67 + 0.448i)12-s + (4.57 + 1.22i)13-s + (2.36 + 4.09i)14-s + (0.500 − 0.866i)16-s + (2.89 − 0.776i)18-s − 3.19i·19-s + (7.09 + 4.09i)21-s + (0.896 − 3.34i)22-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.707 − 0.707i)3-s + (−0.433 + 0.249i)4-s + (−0.353 + 0.612i)6-s + (−1.72 + 0.462i)7-s + (0.249 + 0.249i)8-s + 0.999i·9-s + (0.904 + 0.522i)11-s + (0.482 + 0.129i)12-s + (1.26 + 0.339i)13-s + (0.632 + 1.09i)14-s + (0.125 − 0.216i)16-s + (0.683 − 0.183i)18-s − 0.733i·19-s + (1.54 + 0.894i)21-s + (0.191 − 0.713i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.976 - 0.216i$
motivic weight  =  \(1\)
character  :  $\chi_{450} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 450,\ (\ :1/2),\ 0.976 - 0.216i)\)
\(L(1)\)  \(\approx\)  \(0.661253 + 0.0723370i\)
\(L(\frac12)\)  \(\approx\)  \(0.661253 + 0.0723370i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (4.57 - 1.22i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.57 - 1.22i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 3.19iT - 19T^{2} \)
23 \( 1 + (0.568 - 2.12i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.36 - 9.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0980 + 0.169i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.120 + 0.448i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.55 - 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.79 - 5.79i)T + 53iT^{2} \)
59 \( 1 + (-2.76 - 4.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.43 - 5.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 + (8.66 + 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (16.6 - 4.45i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 + (2.56 - 0.688i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.31526606589014593858954817469, −10.31117057919001475963463952847, −9.360160163040018037625811112480, −8.718759981688084114078770597940, −7.19019157729310725541896406837, −6.49712096621408709692697706236, −5.62000533198866993598066389591, −4.07446016023179885560100458073, −2.88697465306370203521149396570, −1.33834612304026098915903614959, 0.54403632112254412588335189628, 3.54924131744862026884585389901, 4.06345827385389773277076613818, 5.91361181404446206703574087967, 6.07585354254357092567120431626, 7.07804447779873205475815927583, 8.492922840740048863363032117842, 9.406674443585517330830998167287, 9.997924992584597904205782811427, 10.85868828203322396579370724962

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