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Properties

Label 2-450-25.11-c1-0-3
Degree $2$
Conductor $450$
Sign $-0.00868 - 0.999i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.220 + 2.22i)5-s + 1.64·7-s + (0.309 + 0.951i)8-s + (−1.48 − 1.67i)10-s + (−0.232 + 0.169i)11-s + (1.02 + 0.747i)13-s + (−1.32 + 0.964i)14-s + (−0.809 − 0.587i)16-s + (1.52 + 4.68i)17-s + (−0.745 − 2.29i)19-s + (2.18 + 0.478i)20-s + (0.0889 − 0.273i)22-s + (0.588 − 0.427i)23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.0984 + 0.995i)5-s + 0.620·7-s + (0.109 + 0.336i)8-s + (−0.469 − 0.528i)10-s + (−0.0701 + 0.0509i)11-s + (0.285 + 0.207i)13-s + (−0.354 + 0.257i)14-s + (−0.202 − 0.146i)16-s + (0.369 + 1.13i)17-s + (−0.171 − 0.526i)19-s + (0.488 + 0.106i)20-s + (0.0189 − 0.0583i)22-s + (0.122 − 0.0890i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00868 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00868 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.00868 - 0.999i$
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: \((2,\ 450,\ (\ :1/2),\ -0.00868 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771063 + 0.777791i\)
\(L(\frac12)\) \(\approx\) \(0.771063 + 0.777791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.220 - 2.22i)T \)
good7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 + (0.232 - 0.169i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.02 - 0.747i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.52 - 4.68i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.745 + 2.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.588 + 0.427i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.43 - 4.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.27 - 6.99i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.70 - 4.87i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.21 + 1.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.96T + 43T^{2} \)
47 \( 1 + (1.71 - 5.27i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.99 + 6.15i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.03 + 5.11i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.45 - 6.14i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.78 - 11.6i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.76 + 14.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.40 + 3.92i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.98 + 9.18i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.898 + 2.76i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-12.4 + 9.00i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.560 + 1.72i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.94684654048825074700620291067, −10.55894234536765423073725952975, −9.507724488931627636901503099303, −8.485009008229532892773383406073, −7.69071184268750935843910584408, −6.73562993469569253983185697157, −5.95391023759517243628395602369, −4.67330958762107006943891897393, −3.18801876712169083505498960648, −1.69508495601084892940129474930, 0.894938905970605978878204867737, 2.32805683035596422513619578612, 3.94621907042081604082931789684, 5.01484696324065174595600863839, 6.08794447850125109843026361218, 7.65659818169379398228427977617, 8.115718929975835785392275507085, 9.222157334050172099214384456133, 9.756048945166839255365103743726, 10.94294436122447928373441700765

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