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Properties

Label 2-450-15.2-c5-0-18
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 + (22 + 22i)7-s + (45.2 + 45.2i)8-s + 172. i·11-s + (837 − 837i)13-s − 124.·14-s − 256.·16-s + (202. − 202. i)17-s + 1.01e3i·19-s + (−488. − 488. i)22-s + (−2.64e3 − 2.64e3i)23-s + 4.73e3i·26-s + (352. − 352. i)28-s + 2.75e3·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.169 + 0.169i)7-s + (0.250 + 0.250i)8-s + 0.429i·11-s + (1.37 − 1.37i)13-s − 0.169·14-s − 0.250·16-s + (0.169 − 0.169i)17-s + 0.643i·19-s + (−0.214 − 0.214i)22-s + (−1.04 − 1.04i)23-s + 1.37i·26-s + (0.0848 − 0.0848i)28-s + 0.609·29-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: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.415536297\)
\(L(\frac12)\) \(\approx\) \(1.415536297\)
\(L(\frac{7}{2})\) 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 \( 1 + (-22 - 22i)T + 1.68e4iT^{2} \)
11 \( 1 - 172. iT - 1.61e5T^{2} \)
13 \( 1 + (-837 + 837i)T - 3.71e5iT^{2} \)
17 \( 1 + (-202. + 202. i)T - 1.41e6iT^{2} \)
19 \( 1 - 1.01e3iT - 2.47e6T^{2} \)
23 \( 1 + (2.64e3 + 2.64e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 2.75e3T + 2.05e7T^{2} \)
31 \( 1 + 4.13e3T + 2.86e7T^{2} \)
37 \( 1 + (-2.66e3 - 2.66e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.03e4iT - 1.15e8T^{2} \)
43 \( 1 + (-1.05e4 + 1.05e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.56e4 - 1.56e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.34e4 - 1.34e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 2.46e4T + 7.14e8T^{2} \)
61 \( 1 + 1.10e4T + 8.44e8T^{2} \)
67 \( 1 + (2.79e4 + 2.79e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 4.24e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.83e3 - 2.83e3i)T - 2.07e9iT^{2} \)
79 \( 1 + 9.86e4iT - 3.07e9T^{2} \)
83 \( 1 + (1.04e4 + 1.04e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.14e5T + 5.58e9T^{2} \)
97 \( 1 + (2.30e4 + 2.30e4i)T + 8.58e9iT^{2} \)
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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

−10.26023338876403603222986081693, −9.148385953135324763856501433224, −8.231079936690065061462444504925, −7.69134067499974106398551910819, −6.35155913586734101550468587637, −5.73215192043264791731208279688, −4.50735835610405804154421411473, −3.18008341731904019852836399256, −1.69154715045129566082061985657, −0.46955512793426057167669929103, 1.02199109333388778631456112995, 2.02399164838305875873238487621, 3.47870630041908092384692148363, 4.29732757116342234553535463438, 5.77890132954879138630058383062, 6.77375742715120001626337128626, 7.83531594787658533326525979236, 8.763458440683255648732106705271, 9.395123708684965105746830887470, 10.48329858662080457399606829169

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