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 + (−1.22 − 0.328i)7-s + (−0.707 + 0.707i)8-s − 2.99i·9-s + (3 − 1.73i)11-s + (1.67 − 0.448i)12-s + (1.22 − 0.328i)13-s + (−0.633 + 1.09i)14-s + (0.500 + 0.866i)16-s + (−2.89 − 0.776i)18-s − 7.19i·19-s + (1.90 − 1.09i)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 + (−0.462 − 0.124i)7-s + (−0.249 + 0.249i)8-s − 0.999i·9-s + (0.904 − 0.522i)11-s + (0.482 − 0.129i)12-s + (0.339 − 0.0910i)13-s + (−0.169 + 0.293i)14-s + (0.125 + 0.216i)16-s + (−0.683 − 0.183i)18-s − 1.65i·19-s + (0.415 − 0.239i)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  :  $\chi_{450} (293, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 450,\ (\ :1/2),\ -0.0165 + 0.999i)\)
\(L(1)\)  \(\approx\)  \(0.705000 - 0.716775i\)
\(L(\frac12)\)  \(\approx\)  \(0.705000 - 0.716775i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.22 + 0.328i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.22 + 0.328i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 7.19iT - 19T^{2} \)
23 \( 1 + (2.12 + 7.91i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.63 - 6.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.09 + 8.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.55 - 1.55i)T - 37iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.67 + 6.24i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.55 - 5.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.55 + 1.55i)T - 53iT^{2} \)
59 \( 1 + (6.23 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.22 + 12.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (-3.67 - 3.67i)T + 73iT^{2} \)
79 \( 1 + (8.66 - 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.57 - 1.76i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 + (-14.1 - 3.79i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.89480220630815091341822652827, −10.18753586293200805350449927004, −9.245317150708813296284028370765, −8.582403712078537079451003899316, −6.77037037847519040978282736454, −6.09997737642586880709628051401, −4.83529214180051570756799681045, −4.00982937557795730917977902309, −2.86562912005665874930117973554, −0.70357891176458529021146494111, 1.56450438086251025176548040031, 3.54728895740108376623479091716, 4.81081721033639541095872205646, 5.99820663256196502248451597915, 6.46402268720366999939989950864, 7.50975492968959305057196513874, 8.293621432095647500502163683577, 9.527755109533750423582152240058, 10.33902240520984046265059520737, 11.72165090968952540104804107271

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