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Properties

Label 2-450-15.8-c5-0-29
Degree $2$
Conductor $450$
Sign $-0.749 + 0.662i$
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 + (103. − 103. i)7-s + (−45.2 + 45.2i)8-s − 744. i·11-s + (−301. − 301. i)13-s + 582.·14-s − 256.·16-s + (566. + 566. i)17-s − 282. i·19-s + (2.10e3 − 2.10e3i)22-s + (−3.52e3 + 3.52e3i)23-s − 1.70e3i·26-s + (1.64e3 + 1.64e3i)28-s − 6.54e3·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.794 − 0.794i)7-s + (−0.250 + 0.250i)8-s − 1.85i·11-s + (−0.495 − 0.495i)13-s + 0.794·14-s − 0.250·16-s + (0.475 + 0.475i)17-s − 0.179i·19-s + (0.927 − 0.927i)22-s + (−1.38 + 1.38i)23-s − 0.495i·26-s + (0.397 + 0.397i)28-s − 1.44·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(72.1727\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7058474731\)
\(L(\frac12)\) \(\approx\) \(0.7058474731\)
\(L(\frac{7}{2})\) 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 \( 1 + (-103. + 103. i)T - 1.68e4iT^{2} \)
11 \( 1 + 744. iT - 1.61e5T^{2} \)
13 \( 1 + (301. + 301. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-566. - 566. i)T + 1.41e6iT^{2} \)
19 \( 1 + 282. iT - 2.47e6T^{2} \)
23 \( 1 + (3.52e3 - 3.52e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 6.54e3T + 2.05e7T^{2} \)
31 \( 1 + 6.41e3T + 2.86e7T^{2} \)
37 \( 1 + (7.02e3 - 7.02e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 4.01e3iT - 1.15e8T^{2} \)
43 \( 1 + (-4.30e3 - 4.30e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.27e3 + 1.27e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (452. - 452. i)T - 4.18e8iT^{2} \)
59 \( 1 + 1.36e4T + 7.14e8T^{2} \)
61 \( 1 - 2.75e4T + 8.44e8T^{2} \)
67 \( 1 + (2.43e4 - 2.43e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.79e4iT - 1.80e9T^{2} \)
73 \( 1 + (1.38e3 + 1.38e3i)T + 2.07e9iT^{2} \)
79 \( 1 + 5.48e4iT - 3.07e9T^{2} \)
83 \( 1 + (1.94e4 - 1.94e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.41e5T + 5.58e9T^{2} \)
97 \( 1 + (-1.21e5 + 1.21e5i)T - 8.58e9iT^{2} \)
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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

−10.02414963160312644014554796137, −8.762968604166888213299580868021, −7.911464184601796031617569710133, −7.32084868212010825515906291479, −5.90269029137164910488777559257, −5.39044349720747930160845324176, −4.01326091416152494274144412683, −3.27116369579064220284777607491, −1.55855582805331705654976764003, −0.12624230548712318947021804397, 1.83350524718394051788344636848, 2.26781757255347584371174699904, 3.95522451389587382926739598910, 4.82821587727283232120639348465, 5.63160290606316781198106993730, 6.95035276777536999960229126438, 7.81653013441418185118685238220, 9.076271395611067962889182052652, 9.789342209979438706085678423508, 10.69560442880514074728316199501

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