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Properties

Label 2-450-15.8-c5-0-13
Degree $2$
Conductor $450$
Sign $0.374 + 0.927i$
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 + (−148. + 148. i)7-s + (45.2 − 45.2i)8-s − 299. i·11-s + (193. + 193. i)13-s + 841.·14-s − 256.·16-s + (−1.52e3 − 1.52e3i)17-s + 1.74e3i·19-s + (−847. + 847. i)22-s + (−2.53e3 + 2.53e3i)23-s − 1.09e3i·26-s + (−2.37e3 − 2.37e3i)28-s + 2.61e3·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.14 + 1.14i)7-s + (0.250 − 0.250i)8-s − 0.746i·11-s + (0.317 + 0.317i)13-s + 1.14·14-s − 0.250·16-s + (−1.27 − 1.27i)17-s + 1.11i·19-s + (−0.373 + 0.373i)22-s + (−1.00 + 1.00i)23-s − 0.317i·26-s + (−0.573 − 0.573i)28-s + 0.577·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.374 + 0.927i$
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: \((2,\ 450,\ (\ :5/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6970426309\)
\(L(\frac12)\) \(\approx\) \(0.6970426309\)
\(L(\frac{7}{2})\) 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 \( 1 + (148. - 148. i)T - 1.68e4iT^{2} \)
11 \( 1 + 299. iT - 1.61e5T^{2} \)
13 \( 1 + (-193. - 193. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.52e3 + 1.52e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.74e3iT - 2.47e6T^{2} \)
23 \( 1 + (2.53e3 - 2.53e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 2.61e3T + 2.05e7T^{2} \)
31 \( 1 + 8.65e3T + 2.86e7T^{2} \)
37 \( 1 + (-5.37e3 + 5.37e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.58e3iT - 1.15e8T^{2} \)
43 \( 1 + (2.00e3 + 2.00e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.10e4 - 1.10e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (-9.87e3 + 9.87e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 1.39e4T + 7.14e8T^{2} \)
61 \( 1 + 2.81e4T + 8.44e8T^{2} \)
67 \( 1 + (2.41e4 - 2.41e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 6.51e3iT - 1.80e9T^{2} \)
73 \( 1 + (-6.15e4 - 6.15e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 4.24e4iT - 3.07e9T^{2} \)
83 \( 1 + (-3.38e4 + 3.38e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 6.39e4T + 5.58e9T^{2} \)
97 \( 1 + (-5.89e4 + 5.89e4i)T - 8.58e9iT^{2} \)
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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

−9.936909103834191523901768367393, −9.216084457733642095550835589494, −8.671066013467361048252046771179, −7.46434582057695704950662063895, −6.33084247042091802545539907164, −5.55365309765232748650364446631, −3.93680950981461243521722798943, −2.95604425718516036784586563074, −1.95538937773686063225321413526, −0.30905470996921978372044393143, 0.64766646985999600484168286756, 2.17750913046469364878128946265, 3.73461616529228697230722352866, 4.63702269362355388653498670067, 6.19767343400839519359258547257, 6.73103565607275732124577115291, 7.60510710785325937934420257950, 8.687019100617901275862952539329, 9.526817911095968531336375305712, 10.46342020655397728552557927457

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