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Properties

Label 2-450-45.2-c1-0-4
Degree $2$
Conductor $450$
Sign $-0.0165 - 0.999i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0165 - 0.999i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.0165 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705000 + 0.716775i\)
\(L(\frac12)\) \(\approx\) \(0.705000 + 0.716775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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