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Properties

Label 2-450-15.2-c5-0-16
Degree $2$
Conductor $450$
Sign $0.279 + 0.960i$
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  + (2.82 − 2.82i)2-s − 16.0i·4-s + (−50.1 − 50.1i)7-s + (−45.2 − 45.2i)8-s + 659. i·11-s + (517. − 517. i)13-s − 283.·14-s − 256.·16-s + (−62.7 + 62.7i)17-s + 38.1i·19-s + (1.86e3 + 1.86e3i)22-s + (455. + 455. i)23-s − 2.92e3i·26-s + (−802. + 802. i)28-s + 4.59e3·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.386 − 0.386i)7-s + (−0.250 − 0.250i)8-s + 1.64i·11-s + (0.849 − 0.849i)13-s − 0.386·14-s − 0.250·16-s + (−0.0526 + 0.0526i)17-s + 0.0242i·19-s + (0.821 + 0.821i)22-s + (0.179 + 0.179i)23-s − 0.849i·26-s + (−0.193 + 0.193i)28-s + 1.01·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(72.1727\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.674659445\)
\(L(\frac12)\) \(\approx\) \(2.674659445\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 + 2.82i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (50.1 + 50.1i)T + 1.68e4iT^{2} \)
11 \( 1 - 659. iT - 1.61e5T^{2} \)
13 \( 1 + (-517. + 517. i)T - 3.71e5iT^{2} \)
17 \( 1 + (62.7 - 62.7i)T - 1.41e6iT^{2} \)
19 \( 1 - 38.1iT - 2.47e6T^{2} \)
23 \( 1 + (-455. - 455. i)T + 6.43e6iT^{2} \)
29 \( 1 - 4.59e3T + 2.05e7T^{2} \)
31 \( 1 - 2.69e3T + 2.86e7T^{2} \)
37 \( 1 + (-1.48e3 - 1.48e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 194. iT - 1.15e8T^{2} \)
43 \( 1 + (-9.86e3 + 9.86e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-5.30e3 + 5.30e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (2.05e4 + 2.05e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 1.86e4T + 7.14e8T^{2} \)
61 \( 1 + 4.48e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.57e4 - 2.57e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 6.14e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.72e4 + 2.72e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 7.33e4iT - 3.07e9T^{2} \)
83 \( 1 + (-5.89e4 - 5.89e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.12e5T + 5.58e9T^{2} \)
97 \( 1 + (8.79e4 + 8.79e4i)T + 8.58e9iT^{2} \)
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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

−10.20572845440993160524322430516, −9.517026707739922954690390139361, −8.296955944261758218787500655356, −7.19326643360938552544461002082, −6.31367309481035262983326928362, −5.13947778298245463393968239905, −4.20249942803887813348647675264, −3.17051354536169264977282404248, −1.94078259036216266452430301106, −0.67591130455786755124519104060, 0.940569567668271333858021144027, 2.69787054705816065397443405621, 3.63560203237202469638604461451, 4.76317313830161770187728699079, 6.10809982577963363487157692273, 6.31000693344072444734266300976, 7.72550466762344821554655131868, 8.659863609261679317707084070161, 9.251027470351590332372470944788, 10.68432682218208991230445618780

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