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Properties

Label 2-450-25.6-c1-0-12
Degree $2$
Conductor $450$
Sign $-0.596 + 0.802i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−1.24 − 1.85i)5-s − 2.68·7-s + (−0.809 − 0.587i)8-s + (1.37 − 1.75i)10-s + (−0.167 − 0.515i)11-s + (−1.55 + 4.79i)13-s + (−0.829 − 2.55i)14-s + (0.309 − 0.951i)16-s + (−6.22 − 4.52i)17-s + (−5.05 − 3.67i)19-s + (2.09 + 0.768i)20-s + (0.438 − 0.318i)22-s + (−1.50 − 4.64i)23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.557 − 0.829i)5-s − 1.01·7-s + (−0.286 − 0.207i)8-s + (0.436 − 0.556i)10-s + (−0.0504 − 0.155i)11-s + (−0.431 + 1.32i)13-s + (−0.221 − 0.681i)14-s + (0.0772 − 0.237i)16-s + (−1.50 − 1.09i)17-s + (−1.16 − 0.843i)19-s + (0.469 + 0.171i)20-s + (0.0934 − 0.0679i)22-s + (−0.314 − 0.968i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.596 + 0.802i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0970143 - 0.192825i\)
\(L(\frac12)\) \(\approx\) \(0.0970143 - 0.192825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
5 \( 1 + (1.24 + 1.85i)T \)
good7 \( 1 + 2.68T + 7T^{2} \)
11 \( 1 + (0.167 + 0.515i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.55 - 4.79i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (6.22 + 4.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.05 + 3.67i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.50 + 4.64i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.62 + 3.36i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.53 - 1.84i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.07 - 3.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.70 - 5.24i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 + (1.41 - 1.02i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.24 - 5.98i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.245 - 0.756i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.530 + 1.63i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (11.0 + 8.00i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-6.99 + 5.08i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.402 - 1.23i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.61 + 1.90i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.617 - 0.448i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.26 - 6.96i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (9.24 - 6.71i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.86599916116181202155025826199, −9.397294561985220605294877613227, −9.048649909208143190209421559889, −8.046413321144505167988349263277, −6.78835707119555689615315650368, −6.39075547196431933764023657059, −4.68176278801596538248871906807, −4.34749682683520397375369310230, −2.65449124068838371571456844179, −0.11702319349277596179775420394, 2.33030385662199342599418954280, 3.40445286722580809616375407969, 4.26523597004828416398849455023, 5.83662741561414091451730710869, 6.63817302291146173110878115979, 7.80254429443355193091058734761, 8.771302192830032569393032586925, 10.02280587354732225768428047178, 10.50334564852993287114187835119, 11.24771606929469597651223804073

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