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Properties

Label 2-450-25.11-c1-0-6
Degree $2$
Conductor $450$
Sign $0.910 + 0.413i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (1.38 + 1.75i)5-s − 0.447·7-s + (−0.309 − 0.951i)8-s + (2.15 + 0.609i)10-s + (4.42 − 3.21i)11-s + (−0.573 − 0.416i)13-s + (−0.361 + 0.262i)14-s + (−0.809 − 0.587i)16-s + (0.133 + 0.411i)17-s + (2.49 + 7.67i)19-s + (2.09 − 0.771i)20-s + (1.69 − 5.20i)22-s + (4.65 − 3.37i)23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.618 + 0.786i)5-s − 0.169·7-s + (−0.109 − 0.336i)8-s + (0.680 + 0.192i)10-s + (1.33 − 0.970i)11-s + (−0.159 − 0.115i)13-s + (−0.0967 + 0.0702i)14-s + (−0.202 − 0.146i)16-s + (0.0324 + 0.0997i)17-s + (0.571 + 1.76i)19-s + (0.469 − 0.172i)20-s + (0.360 − 1.10i)22-s + (0.969 − 0.704i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11262 - 0.457285i\)
\(L(\frac12)\) \(\approx\) \(2.11262 - 0.457285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
5 \( 1 + (-1.38 - 1.75i)T \)
good7 \( 1 + 0.447T + 7T^{2} \)
11 \( 1 + (-4.42 + 3.21i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.573 + 0.416i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.133 - 0.411i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.49 - 7.67i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.65 + 3.37i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.03 + 3.17i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.407 + 1.25i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.14 + 5.91i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.80 + 3.48i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.09T + 43T^{2} \)
47 \( 1 + (2.83 - 8.73i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.210 - 0.648i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.75 - 2.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.91 + 3.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.528 - 1.62i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.558 + 1.71i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (10.0 - 7.33i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.72 - 8.37i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.545 + 1.67i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (11.2 - 8.18i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.04 + 12.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.10240234290237586814120222788, −10.23513641587685830960029901153, −9.504453315886064697556531120907, −8.417635392760737964606934738344, −7.00469792388741126222686136611, −6.22548189217583401823920251842, −5.43828532323250386409106436758, −3.86968951995979710772997184089, −3.07674319760653418226394004779, −1.56857866548950018080607151306, 1.60561639823574283710200319340, 3.26085167102015269089436842894, 4.68916627542467161664057121776, 5.18800162730189328233888527122, 6.64505736192433724983320686861, 7.05962933545937125274794110361, 8.589020558408213973507910033264, 9.246210935040650846623088596035, 10.04218709869165987329171726998, 11.52989262835958131274706208914

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