Properties

Label 2-450-15.8-c5-0-11
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 + (52 − 52i)7-s + (−45.2 + 45.2i)8-s − 124. i·11-s + (−183 − 183i)13-s + 294.·14-s − 256.·16-s + (−350. − 350. i)17-s + 2.10e3i·19-s + (352 − 352i)22-s + (−1.81e3 + 1.81e3i)23-s − 1.03e3i·26-s + (832. + 832. i)28-s + 6.99e3·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.401 − 0.401i)7-s + (−0.250 + 0.250i)8-s − 0.310i·11-s + (−0.300 − 0.300i)13-s + 0.401·14-s − 0.250·16-s + (−0.294 − 0.294i)17-s + 1.33i·19-s + (0.155 − 0.155i)22-s + (−0.714 + 0.714i)23-s − 0.300i·26-s + (0.200 + 0.200i)28-s + 1.54·29-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

\(L(3)\) \(\approx\) \(2.334905069\)
\(L(\frac12)\) \(\approx\) \(2.334905069\)
\(L(\frac{7}{2})\) 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 \( 1 + (-52 + 52i)T - 1.68e4iT^{2} \)
11 \( 1 + 124. iT - 1.61e5T^{2} \)
13 \( 1 + (183 + 183i)T + 3.71e5iT^{2} \)
17 \( 1 + (350. + 350. i)T + 1.41e6iT^{2} \)
19 \( 1 - 2.10e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.81e3 - 1.81e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 6.99e3T + 2.05e7T^{2} \)
31 \( 1 - 6.78e3T + 2.86e7T^{2} \)
37 \( 1 + (3.72e3 - 3.72e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.22e4iT - 1.15e8T^{2} \)
43 \( 1 + (3.18e3 + 3.18e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-7.76e3 - 7.76e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (-2.28e4 + 2.28e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 3.58e4T + 7.14e8T^{2} \)
61 \( 1 - 1.37e4T + 8.44e8T^{2} \)
67 \( 1 + (3.40e4 - 3.40e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 4.80e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.98e4 - 2.98e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 5.93e4iT - 3.07e9T^{2} \)
83 \( 1 + (-3.81e4 + 3.81e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.70e4T + 5.58e9T^{2} \)
97 \( 1 + (9.60e4 - 9.60e4i)T - 8.58e9iT^{2} \)
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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

−10.52180693712923897713075082023, −9.753280680754488838339892845319, −8.388925650301261308207054974352, −7.87841427831711052170750743566, −6.78046056911968728777764182727, −5.87198585669582634560332588210, −4.84865497172954611395228692688, −3.90514324453486619891727921943, −2.71508134970827262013081669291, −1.16778263234508688776017036118, 0.48979524782549464151845284154, 1.95634958149178792469295721614, 2.83279513606061245539069270527, 4.29773005788543456470195232346, 4.95237443059099131570385810800, 6.16522217850273999102494474260, 7.06567964675805913195739350332, 8.358106568698121681076734423305, 9.147398598187645585782443696141, 10.22956715341008914072327410377

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