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Properties

Label 2-450-45.4-c1-0-1
Degree $2$
Conductor $450$
Sign $-0.974 - 0.223i$
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  + (−0.866 + 0.5i)2-s + (−1.41 + i)3-s + (0.499 − 0.866i)4-s + (0.724 − 1.57i)6-s + (−0.389 + 0.224i)7-s + 0.999i·8-s + (1.00 − 2.82i)9-s + (2.44 + 4.24i)11-s + (0.158 + 1.72i)12-s + (−0.389 − 0.224i)13-s + (0.224 − 0.389i)14-s + (−0.5 − 0.866i)16-s + 4.89i·17-s + (0.548 + 2.94i)18-s − 7.44·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.816 + 0.577i)3-s + (0.249 − 0.433i)4-s + (0.295 − 0.642i)6-s + (−0.147 + 0.0849i)7-s + 0.353i·8-s + (0.333 − 0.942i)9-s + (0.738 + 1.27i)11-s + (0.0458 + 0.497i)12-s + (−0.107 − 0.0623i)13-s + (0.0600 − 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.18i·17-s + (0.129 + 0.695i)18-s − 1.70·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.974 - 0.223i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0494193 + 0.436928i\)
\(L(\frac12)\) \(\approx\) \(0.0494193 + 0.436928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 \)
good7 \( 1 + (0.389 - 0.224i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.389 + 0.224i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (2.12 + 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.43 - 5.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.55iT - 53T^{2} \)
59 \( 1 + (2.72 - 4.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.301 + 0.174i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + (-8.34 - 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.71 + 2.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-7.61 + 4.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.20964086534093093719395289903, −10.52247006148435891280839425252, −9.741748894179526882789591052833, −9.002173935329932866075159043522, −7.88287515744256140888478670249, −6.62423357649959780708051378830, −6.18098034586474840614404667839, −4.82350974321019474402189743799, −3.92721899495613422447558742065, −1.81270793279332924964849194958, 0.36335831074920980610730063109, 1.93547569987324542925252039391, 3.51645064183507957150406719971, 4.97144804136592502409832359654, 6.25453314511661988134404561203, 6.84157883627047920079896088629, 8.028655328214989937270892775076, 8.815269618297406824661426997502, 9.925806007678699233605115497240, 10.81601058615324388111884495892

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