Properties

Label 2-450-5.4-c7-0-21
Degree $2$
Conductor $450$
Sign $0.447 - 0.894i$
Analytic cond. $140.573$
Root an. cond. $11.8563$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − 64·4-s + 1.17e3i·7-s + 512i·8-s + 7.56e3·11-s + 5.37e3i·13-s + 9.39e3·14-s + 4.09e3·16-s + 2.40e4i·17-s + 5.12e4·19-s − 6.05e4i·22-s + 5.76e4i·23-s + 4.29e4·26-s − 7.51e4i·28-s + 4.70e4·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.29i·7-s + 0.353i·8-s + 1.71·11-s + 0.678i·13-s + 0.914·14-s + 0.250·16-s + 1.18i·17-s + 1.71·19-s − 1.21i·22-s + 0.987i·23-s + 0.479·26-s − 0.646i·28-s + 0.358·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(140.573\)
Root analytic conductor: \(11.8563\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :7/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.308117870\)
\(L(\frac12)\) \(\approx\) \(2.308117870\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.17e3iT - 8.23e5T^{2} \)
11 \( 1 - 7.56e3T + 1.94e7T^{2} \)
13 \( 1 - 5.37e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.40e4iT - 4.10e8T^{2} \)
19 \( 1 - 5.12e4T + 8.93e8T^{2} \)
23 \( 1 - 5.76e4iT - 3.40e9T^{2} \)
29 \( 1 - 4.70e4T + 1.72e10T^{2} \)
31 \( 1 + 1.92e5T + 2.75e10T^{2} \)
37 \( 1 + 1.97e5iT - 9.49e10T^{2} \)
41 \( 1 - 2.37e5T + 1.94e11T^{2} \)
43 \( 1 - 6.53e5iT - 2.71e11T^{2} \)
47 \( 1 + 8.26e5iT - 5.06e11T^{2} \)
53 \( 1 + 5.69e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.50e6T + 2.48e12T^{2} \)
61 \( 1 + 2.06e6T + 3.14e12T^{2} \)
67 \( 1 - 3.44e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.12e6T + 9.09e12T^{2} \)
73 \( 1 + 8.36e4iT - 1.10e13T^{2} \)
79 \( 1 + 1.45e6T + 1.92e13T^{2} \)
83 \( 1 + 1.62e6iT - 2.71e13T^{2} \)
89 \( 1 - 6.00e6T + 4.42e13T^{2} \)
97 \( 1 + 3.41e6iT - 8.07e13T^{2} \)
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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

−9.923374914748787078762963026552, −9.187059666983558371329088601495, −8.723000558343305132376587159244, −7.39465321681355895676386792606, −6.18286103332631702848611466223, −5.39665676722029548517522317807, −4.08770703395659494188646269971, −3.25826170708675744754859853191, −1.93555770967744663444818122795, −1.21917058282298741209046473426, 0.53036075983103706953905413070, 1.21708766344438433398644374452, 3.16391172735101543961543113472, 4.08569328755746368262881645946, 5.02765887727846932244141304907, 6.21451142370529435875862578363, 7.14600376422219506256592127201, 7.60748434435774759863021983543, 8.923446492308061568783926699954, 9.598125240927485583771544026269

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