Properties

Degree $2$
Conductor $450$
Sign $-0.894 + 0.447i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 16·4-s + 118i·7-s − 64i·8-s − 192·11-s + 1.10e3i·13-s − 472·14-s + 256·16-s + 762i·17-s + 2.74e3·19-s − 768i·22-s − 1.56e3i·23-s − 4.42e3·26-s − 1.88e3i·28-s + 5.91e3·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.910i·7-s − 0.353i·8-s − 0.478·11-s + 1.81i·13-s − 0.643·14-s + 0.250·16-s + 0.639i·17-s + 1.74·19-s − 0.338i·22-s − 0.617i·23-s − 1.28·26-s − 0.455i·28-s + 1.30·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Motivic weight: \(5\)
Character: $\chi_{450} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.195334073\)
\(L(\frac12)\) \(\approx\) \(1.195334073\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 118iT - 1.68e4T^{2} \)
11 \( 1 + 192T + 1.61e5T^{2} \)
13 \( 1 - 1.10e3iT - 3.71e5T^{2} \)
17 \( 1 - 762iT - 1.41e6T^{2} \)
19 \( 1 - 2.74e3T + 2.47e6T^{2} \)
23 \( 1 + 1.56e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.91e3T + 2.05e7T^{2} \)
31 \( 1 + 6.86e3T + 2.86e7T^{2} \)
37 \( 1 - 5.51e3iT - 6.93e7T^{2} \)
41 \( 1 - 378T + 1.15e8T^{2} \)
43 \( 1 + 2.43e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.31e4iT - 2.29e8T^{2} \)
53 \( 1 - 9.17e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.49e4T + 7.14e8T^{2} \)
61 \( 1 + 9.83e3T + 8.44e8T^{2} \)
67 \( 1 + 3.37e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.02e4T + 1.80e9T^{2} \)
73 \( 1 - 2.19e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.52e3T + 3.07e9T^{2} \)
83 \( 1 - 1.09e5iT - 3.93e9T^{2} \)
89 \( 1 - 3.84e4T + 5.58e9T^{2} \)
97 \( 1 - 1.91e3iT - 8.58e9T^{2} \)
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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

−10.76656733887670664673149642774, −9.557455293700342535705976263121, −9.004276915383647133073234122526, −8.065596996378590770652163928973, −7.05922621110082968527470876357, −6.17541032039696109384632373287, −5.24077279646477479109744971849, −4.26548087935373380214980896378, −2.86157357112046419438571410788, −1.48842805208648658581262739744, 0.31281360253494608468916559672, 1.18624832241486541605750409412, 2.83459954782321005906137828440, 3.55652948421125300300334901107, 4.92878701657058044852398638875, 5.67673421656492059811411036886, 7.33386643894994979579433650161, 7.81100620797478577951178630402, 9.061787114554650116522998804702, 10.08241192306380424153451880554

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