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Properties

Label 2-450-15.2-c5-0-17
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 + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

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

\(L(3)\) \(\approx\) \(1.258607535\)
\(L(\frac12)\) \(\approx\) \(1.258607535\)
\(L(\frac{7}{2})\) 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

−9.925318831871462542118813522997, −9.220327553319829771084656511434, −8.392211531963185701553880764885, −7.45228828370377403276861651904, −6.60749856914098908017679712495, −5.47181328543383787671046626957, −4.67515670771307342481076212185, −3.08465899103667496539752394640, −1.76433654560856909683308097588, −0.41831385284723725214229965921, 0.980505314877866470825167621684, 2.07054086806778998418539445143, 3.38811373625323384211773989357, 4.43011102914484687437254467215, 5.62502955408275315206639871243, 6.92780746746238174109714844667, 7.81950697743832316501161196635, 8.556624794426254278876342779801, 9.685242833924921719613398237469, 10.35801984115839821530040346616

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