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Properties

Label 2-450-15.8-c5-0-26
Degree $2$
Conductor $450$
Sign $0.884 + 0.465i$
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 + (158. − 158. i)7-s + (−45.2 + 45.2i)8-s − 147. i·11-s + (516. + 516. i)13-s + 897.·14-s − 256.·16-s + (−1.09e3 − 1.09e3i)17-s + 1.17e3i·19-s + (415. − 415. i)22-s + (3.13e3 − 3.13e3i)23-s + 2.92e3i·26-s + (2.53e3 + 2.53e3i)28-s − 1.86e3·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (1.22 − 1.22i)7-s + (−0.250 + 0.250i)8-s − 0.366i·11-s + (0.847 + 0.847i)13-s + 1.22·14-s − 0.250·16-s + (−0.917 − 0.917i)17-s + 0.747i·19-s + (0.183 − 0.183i)22-s + (1.23 − 1.23i)23-s + 0.847i·26-s + (0.611 + 0.611i)28-s − 0.411·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.884 + 0.465i$
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: \((2,\ 450,\ (\ :5/2),\ 0.884 + 0.465i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.152804838\)
\(L(\frac12)\) \(\approx\) \(3.152804838\)
\(L(\frac{7}{2})\) 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 \( 1 + (-158. + 158. i)T - 1.68e4iT^{2} \)
11 \( 1 + 147. iT - 1.61e5T^{2} \)
13 \( 1 + (-516. - 516. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.09e3 + 1.09e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 + (-3.13e3 + 3.13e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 1.86e3T + 2.05e7T^{2} \)
31 \( 1 + 3.28e3T + 2.86e7T^{2} \)
37 \( 1 + (-1.68e3 + 1.68e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 9.09e3iT - 1.15e8T^{2} \)
43 \( 1 + (1.37e4 + 1.37e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (-887. - 887. i)T + 2.29e8iT^{2} \)
53 \( 1 + (-1.63e4 + 1.63e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 957.T + 7.14e8T^{2} \)
61 \( 1 - 1.93e4T + 8.44e8T^{2} \)
67 \( 1 + (3.57e4 - 3.57e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 2.59e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.28e4 - 5.28e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 1.63e4iT - 3.07e9T^{2} \)
83 \( 1 + (-6.95e4 + 6.95e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.33e5T + 5.58e9T^{2} \)
97 \( 1 + (8.35e3 - 8.35e3i)T - 8.58e9iT^{2} \)
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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

−10.50686523560639852612474919392, −9.006611126892477786912618483034, −8.332405048372863314693438058268, −7.25175673515586367307369092431, −6.67517947194489969161739447937, −5.31997600605902836498862808974, −4.44471015112227841581886019703, −3.65459086260920463831922560594, −1.98843174937285472882003415445, −0.66019374126202243136758812408, 1.25731669275470770678071483568, 2.19338459952226065204815971342, 3.36050094654247099037978006926, 4.70396863384038619892003666494, 5.39583743869675743224615077228, 6.35532476897146512264542218042, 7.74963596935502790461046713968, 8.669765197226058919846868133651, 9.375575485345359088044328158119, 10.78002715778242702425487768071

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