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Properties

Label 2-450-15.8-c5-0-5
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 + (−140. + 140. i)7-s + (45.2 − 45.2i)8-s + 540. i·11-s + (798. + 798. i)13-s + 795.·14-s − 256.·16-s + (−598. − 598. i)17-s + 1.78e3i·19-s + (1.53e3 − 1.53e3i)22-s + (718. − 718. i)23-s − 4.51e3i·26-s + (−2.25e3 − 2.25e3i)28-s + 5.38e3·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.08 + 1.08i)7-s + (0.250 − 0.250i)8-s + 1.34i·11-s + (1.31 + 1.31i)13-s + 1.08·14-s − 0.250·16-s + (−0.502 − 0.502i)17-s + 1.13i·19-s + (0.673 − 0.673i)22-s + (0.283 − 0.283i)23-s − 1.31i·26-s + (−0.542 − 0.542i)28-s + 1.18·29-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

\(L(3)\) \(\approx\) \(0.9180338959\)
\(L(\frac12)\) \(\approx\) \(0.9180338959\)
\(L(\frac{7}{2})\) 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 \( 1 + (140. - 140. i)T - 1.68e4iT^{2} \)
11 \( 1 - 540. iT - 1.61e5T^{2} \)
13 \( 1 + (-798. - 798. i)T + 3.71e5iT^{2} \)
17 \( 1 + (598. + 598. i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.78e3iT - 2.47e6T^{2} \)
23 \( 1 + (-718. + 718. i)T - 6.43e6iT^{2} \)
29 \( 1 - 5.38e3T + 2.05e7T^{2} \)
31 \( 1 - 1.80e3T + 2.86e7T^{2} \)
37 \( 1 + (7.34e3 - 7.34e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.24e4iT - 1.15e8T^{2} \)
43 \( 1 + (9.90e3 + 9.90e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.82e3 - 1.82e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.60e4 - 1.60e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 4.52e4T + 7.14e8T^{2} \)
61 \( 1 - 605.T + 8.44e8T^{2} \)
67 \( 1 + (-2.30e4 + 2.30e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 1.13e4iT - 1.80e9T^{2} \)
73 \( 1 + (4.18e4 + 4.18e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 6.19e4iT - 3.07e9T^{2} \)
83 \( 1 + (-8.21e4 + 8.21e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.07e4T + 5.58e9T^{2} \)
97 \( 1 + (5.14e4 - 5.14e4i)T - 8.58e9iT^{2} \)
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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

−10.48802638404263753559303559036, −9.688761796228289742699369363524, −9.040372226360096431733870507696, −8.259232311761498628408548682971, −6.81991251163686021372329968047, −6.30218101244958129506095066653, −4.78698531801503638184897573225, −3.62987817027949138762437068127, −2.47970395733039532441560846500, −1.46987581032186094731302514522, 0.32316000294710738314574106270, 0.959823902146227938547293391410, 3.01995199289772974526474122609, 3.86331271875821404011673448639, 5.42616693081393225164401051443, 6.35251117522768980145206235566, 7.00452322700879224366461087458, 8.255055004196948960607460880722, 8.774957931322207109281809603290, 9.972351293236235721118353376364

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