Properties

Degree $2$
Conductor $450$
Sign $-0.875 + 0.482i$
Motivic weight $1$
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 + (−2.02 − 0.951i)5-s − 2.77·7-s + (−0.309 − 0.951i)8-s + (−2.19 + 0.420i)10-s + (2.24 − 1.63i)11-s + (−4.59 − 3.33i)13-s + (−2.24 + 1.63i)14-s + (−0.809 − 0.587i)16-s + (−1.59 − 4.90i)17-s + (0.436 + 1.34i)19-s + (−1.52 + 1.63i)20-s + (0.857 − 2.63i)22-s + (−0.529 + 0.384i)23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.905 − 0.425i)5-s − 1.04·7-s + (−0.109 − 0.336i)8-s + (−0.694 + 0.132i)10-s + (0.676 − 0.491i)11-s + (−1.27 − 0.925i)13-s + (−0.599 + 0.435i)14-s + (−0.202 − 0.146i)16-s + (−0.386 − 1.18i)17-s + (0.100 + 0.308i)19-s + (−0.342 + 0.364i)20-s + (0.182 − 0.562i)22-s + (−0.110 + 0.0802i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.875 + 0.482i$
Motivic weight: \(1\)
Character: $\chi_{450} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247232 - 0.960907i\)
\(L(\frac12)\) \(\approx\) \(0.247232 - 0.960907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 + 0.951i)T \)
good7 \( 1 + 2.77T + 7T^{2} \)
11 \( 1 + (-2.24 + 1.63i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (4.59 + 3.33i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.59 + 4.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.436 - 1.34i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.529 - 0.384i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.26 - 3.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.20 + 6.77i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.847 - 0.615i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.36 - 5.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + (-0.857 + 2.63i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.162 + 0.500i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.05 + 2.22i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.76 + 6.36i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.33 + 4.11i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.09 + 12.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.40 - 2.47i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.05 - 9.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.44 + 4.44i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.43 - 5.39i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.0278 - 0.0857i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.97033026352643164553380777010, −9.737573979991978122605716917927, −9.204079835900357708620484278513, −7.83311800814808100095789027938, −6.99363403753537531890478581923, −5.81683684891591967729883129089, −4.76039705733914518911560515898, −3.68082278974410415799941389105, −2.74390655219393818171254703756, −0.49629825910657292722659924570, 2.50370634293060288506765605087, 3.82108622464956884441779126716, 4.47085221419100303420137552249, 6.00797122070845264451479371184, 6.92309172818933738974662449837, 7.40172343944270969959665903591, 8.721321694719491729629119237133, 9.608327045928573240535466888024, 10.67621405236657366543453089101, 11.70988396875957706831376813672

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