Properties

Label 2-450-15.8-c5-0-1
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 + (9.67 − 9.67i)7-s + (−45.2 + 45.2i)8-s + 479. i·11-s + (177. + 177. i)13-s + 54.7·14-s − 256.·16-s + (−1.38e3 − 1.38e3i)17-s − 121. i·19-s + (−1.35e3 + 1.35e3i)22-s + (−1.69e3 + 1.69e3i)23-s + 1.00e3i·26-s + (154. + 154. i)28-s + 7.04e3·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.0746 − 0.0746i)7-s + (−0.250 + 0.250i)8-s + 1.19i·11-s + (0.291 + 0.291i)13-s + 0.0746·14-s − 0.250·16-s + (−1.15 − 1.15i)17-s − 0.0770i·19-s + (−0.597 + 0.597i)22-s + (−0.668 + 0.668i)23-s + 0.291i·26-s + (0.0373 + 0.0373i)28-s + 1.55·29-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

\(L(3)\) \(\approx\) \(0.4521014794\)
\(L(\frac12)\) \(\approx\) \(0.4521014794\)
\(L(\frac{7}{2})\) 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 \( 1 + (-9.67 + 9.67i)T - 1.68e4iT^{2} \)
11 \( 1 - 479. iT - 1.61e5T^{2} \)
13 \( 1 + (-177. - 177. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.38e3 + 1.38e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 121. iT - 2.47e6T^{2} \)
23 \( 1 + (1.69e3 - 1.69e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 7.04e3T + 2.05e7T^{2} \)
31 \( 1 - 443.T + 2.86e7T^{2} \)
37 \( 1 + (4.79e3 - 4.79e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.02e3iT - 1.15e8T^{2} \)
43 \( 1 + (6.31e3 + 6.31e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.28e4 + 1.28e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.67e4 - 1.67e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 4.13e4T + 7.14e8T^{2} \)
61 \( 1 + 3.57e4T + 8.44e8T^{2} \)
67 \( 1 + (-4.01e4 + 4.01e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 3.29e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.03e4 + 2.03e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 2.71e4iT - 3.07e9T^{2} \)
83 \( 1 + (3.22e4 - 3.22e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 6.48e4T + 5.58e9T^{2} \)
97 \( 1 + (2.23e4 - 2.23e4i)T - 8.58e9iT^{2} \)
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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

−10.94052052518482430684741507179, −9.827921251076986823383087010820, −9.001237353567746637201234641820, −7.927680999931675207080219930175, −7.02083606071094434218803950955, −6.32403821662119655840177923433, −4.95041623941168913987993768935, −4.38803168736173290101923984635, −3.00333189522484163680232062155, −1.75225578686905416686050102301, 0.084253886650862377925148214188, 1.42634111648411637478629161953, 2.70424971341060759690028632107, 3.75323492826083905540578240017, 4.75876883105881206819087703597, 5.97535119584480710339578730017, 6.56511490220135023122742454818, 8.195422003481115959717712620738, 8.705965091595995993992420679136, 9.999946238273402050886956420695

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