Properties

Label 2-450-5.4-c5-0-28
Degree $2$
Conductor $450$
Sign $-0.447 + 0.894i$
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 − 233i·7-s + 64i·8-s + 498·11-s + 809i·13-s − 932·14-s + 256·16-s − 1.00e3i·17-s + 1.70e3·19-s − 1.99e3i·22-s − 1.55e3i·23-s + 3.23e3·26-s + 3.72e3i·28-s + 7.83e3·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 1.79i·7-s + 0.353i·8-s + 1.24·11-s + 1.32i·13-s − 1.27·14-s + 0.250·16-s − 0.840i·17-s + 1.08·19-s − 0.877i·22-s − 0.612i·23-s + 0.938·26-s + 0.898i·28-s + 1.72·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(72.1727\)
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: \((2,\ 450,\ (\ :5/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.284518836\)
\(L(\frac12)\) \(\approx\) \(2.284518836\)
\(L(\frac{7}{2})\) 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 \( 1 + 233iT - 1.68e4T^{2} \)
11 \( 1 - 498T + 1.61e5T^{2} \)
13 \( 1 - 809iT - 3.71e5T^{2} \)
17 \( 1 + 1.00e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.70e3T + 2.47e6T^{2} \)
23 \( 1 + 1.55e3iT - 6.43e6T^{2} \)
29 \( 1 - 7.83e3T + 2.05e7T^{2} \)
31 \( 1 - 977T + 2.86e7T^{2} \)
37 \( 1 - 4.82e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.14e3T + 1.15e8T^{2} \)
43 \( 1 - 1.94e4iT - 1.47e8T^{2} \)
47 \( 1 - 8.41e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.76e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.59e4T + 7.14e8T^{2} \)
61 \( 1 - 3.52e3T + 8.44e8T^{2} \)
67 \( 1 + 5.74e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.54e3T + 1.80e9T^{2} \)
73 \( 1 + 646iT - 2.07e9T^{2} \)
79 \( 1 - 2.27e4T + 3.07e9T^{2} \)
83 \( 1 + 1.15e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.89e4T + 5.58e9T^{2} \)
97 \( 1 + 5.45e4iT - 8.58e9T^{2} \)
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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

−9.894035240479646785678045928823, −9.481658581628181040781009738857, −8.266681119550813182480584871326, −7.10403391905505918909641437101, −6.51142085319768684907579093637, −4.67852629915443502086391780684, −4.16231082955827487946164769186, −3.05843028153110295655057029766, −1.42312071288491152235082666376, −0.70177569216353709191054009788, 1.05999910540025849415515964428, 2.60053570576630696783709110047, 3.77961938066780321635110940594, 5.28204674117611898410969703455, 5.78559822479078762602279715531, 6.74461202560975420272404410661, 7.982534451473953299366396006029, 8.707883497582276091591723809566, 9.394012638392848689329039917442, 10.39327360418251247698837314349

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