Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} $
Sign $-0.0165 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.0165 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{450} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 450,\ (\ :1/2),\ -0.0165 - 0.999i)\)
\(L(1)\)  \(\approx\)  \(1.46759 + 1.49210i\)
\(L(\frac12)\)  \(\approx\)  \(1.46759 + 1.49210i\)
\(L(\frac{3}{2})\)   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

−11.17937989934276943719940923759, −10.35503681788829078115036123598, −9.280152992672695818828465726370, −8.585532125675511165256131890731, −7.60460399182183613629919606046, −7.04256002352578485527742892980, −5.12879152080455815691254130661, −4.79173147572037409556315604795, −3.67261656887416147381458942905, −1.98384940408493597502087271947, 1.46937538518598516504100010522, 2.34219956549591211269832497673, 3.79358616511846554887178833585, 4.90763478506545822171899918589, 6.08609573575614039064661733022, 7.46763585117161240333642354073, 8.183143663442704117285343471055, 9.032883206886816496773840878024, 9.855687975344600843879962894673, 11.24454192616127247310135185849

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