Properties

Label 2-450-5.4-c5-0-4
Degree $2$
Conductor $450$
Sign $-0.894 - 0.447i$
Analytic cond. $72.1727$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational no
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 − 148i·7-s − 64i·8-s − 384·11-s + 334i·13-s + 592·14-s + 256·16-s − 576i·17-s + 664·19-s − 1.53e3i·22-s − 3.84e3i·23-s − 1.33e3·26-s + 2.36e3i·28-s + 96·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.14i·7-s − 0.353i·8-s − 0.956·11-s + 0.548i·13-s + 0.807·14-s + 0.250·16-s − 0.483i·17-s + 0.421·19-s − 0.676i·22-s − 1.51i·23-s − 0.387·26-s + 0.570i·28-s + 0.0211·29-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: $-0.894 - 0.447i$
Analytic conductor: \(72.1727\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
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

\(L(3)\) \(\approx\) \(0.6150953627\)
\(L(\frac12)\) \(\approx\) \(0.6150953627\)
\(L(\frac{7}{2})\) 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 \( 1 + 148iT - 1.68e4T^{2} \)
11 \( 1 + 384T + 1.61e5T^{2} \)
13 \( 1 - 334iT - 3.71e5T^{2} \)
17 \( 1 + 576iT - 1.41e6T^{2} \)
19 \( 1 - 664T + 2.47e6T^{2} \)
23 \( 1 + 3.84e3iT - 6.43e6T^{2} \)
29 \( 1 - 96T + 2.05e7T^{2} \)
31 \( 1 + 4.56e3T + 2.86e7T^{2} \)
37 \( 1 - 5.79e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.72e3T + 1.15e8T^{2} \)
43 \( 1 - 1.48e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.92e4iT - 2.29e8T^{2} \)
53 \( 1 - 7.77e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.30e4T + 7.14e8T^{2} \)
61 \( 1 - 4.27e4T + 8.44e8T^{2} \)
67 \( 1 - 3.66e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.45e4T + 1.80e9T^{2} \)
73 \( 1 - 1.68e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.80e4T + 3.07e9T^{2} \)
83 \( 1 + 6.64e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.17e4T + 5.58e9T^{2} \)
97 \( 1 + 2.99e4iT - 8.58e9T^{2} \)
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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

−10.52996242230437359961534617030, −9.778194363185077727071770066547, −8.722315284865624994013131989467, −7.72498097277686902083697945734, −7.11077569551163690015173916383, −6.12051843623748743059620516990, −4.91483544490554812025275985515, −4.14503645007329224063603529960, −2.76456391701795213468103585635, −1.02641140103927649285006683191, 0.16454095383701409999386283611, 1.74998504519263957003663357000, 2.73533929142758571165712593847, 3.75842895960147398168167904966, 5.31571790302174796653268452805, 5.66568585321206979654863011203, 7.31737471736761941968561493578, 8.270360642233371814865417229996, 9.105032260247402833706613158926, 9.965096712655577045952845939012

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