Properties

Label 2-450-15.2-c5-0-10
Degree $2$
Conductor $450$
Sign $0.812 - 0.583i$
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 + (117. + 117. i)7-s + (−45.2 − 45.2i)8-s + 28.2i·11-s + (283. − 283. i)13-s + 662.·14-s − 256.·16-s + (−137. + 137. i)17-s + 2.80e3i·19-s + (79.9 + 79.9i)22-s + (−902. − 902. i)23-s − 1.60e3i·26-s + (1.87e3 − 1.87e3i)28-s − 827.·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.903 + 0.903i)7-s + (−0.250 − 0.250i)8-s + 0.0704i·11-s + (0.464 − 0.464i)13-s + 0.903·14-s − 0.250·16-s + (−0.115 + 0.115i)17-s + 1.78i·19-s + (0.0352 + 0.0352i)22-s + (−0.355 − 0.355i)23-s − 0.464i·26-s + (0.451 − 0.451i)28-s − 0.182·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.812 - 0.583i$
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: \((2,\ 450,\ (\ :5/2),\ 0.812 - 0.583i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.785731383\)
\(L(\frac12)\) \(\approx\) \(2.785731383\)
\(L(\frac{7}{2})\) 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 \( 1 + (-117. - 117. i)T + 1.68e4iT^{2} \)
11 \( 1 - 28.2iT - 1.61e5T^{2} \)
13 \( 1 + (-283. + 283. i)T - 3.71e5iT^{2} \)
17 \( 1 + (137. - 137. i)T - 1.41e6iT^{2} \)
19 \( 1 - 2.80e3iT - 2.47e6T^{2} \)
23 \( 1 + (902. + 902. i)T + 6.43e6iT^{2} \)
29 \( 1 + 827.T + 2.05e7T^{2} \)
31 \( 1 - 2.04e3T + 2.86e7T^{2} \)
37 \( 1 + (1.32e3 + 1.32e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.07e4iT - 1.15e8T^{2} \)
43 \( 1 + (7.83e3 - 7.83e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.17e4 - 1.17e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.25e4 - 2.25e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 3.31e4T + 7.14e8T^{2} \)
61 \( 1 - 3.20e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.74e4 - 2.74e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 5.46e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.42e4 + 1.42e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 7.92e4iT - 3.07e9T^{2} \)
83 \( 1 + (-3.65e4 - 3.65e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 6.99e4T + 5.58e9T^{2} \)
97 \( 1 + (-6.93e4 - 6.93e4i)T + 8.58e9iT^{2} \)
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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

−10.52308389139798042302657030596, −9.643892377199913494212614141604, −8.484026644863968036403338050787, −7.87509033228796531842860589951, −6.29875380511372069098428514663, −5.57079064002882695917974487695, −4.58571919589077194411074549567, −3.45044364606795019870937483357, −2.21718892322195690087519913446, −1.25366592607995799248318913360, 0.57190479825520212918183048444, 2.02234868752179117218272074664, 3.53439811493672210751076336367, 4.50755013632540649148663228833, 5.28346878729467485170293515430, 6.62286638969943723569664894506, 7.26471289563970018510286696167, 8.254796423121522588864062987752, 9.082061456047929432950677598052, 10.33517403182305334299687132062

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