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Properties

Label 2-450-9.2-c2-0-15
Degree $2$
Conductor $450$
Sign $0.559 + 0.828i$
Analytic cond. $12.2616$
Root an. cond. $3.50165$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (−2.36 − 1.84i)3-s + (0.999 + 1.73i)4-s + (1.59 + 3.93i)6-s + (4.79 − 8.30i)7-s − 2.82i·8-s + (2.18 + 8.73i)9-s + (8.83 + 5.10i)11-s + (0.833 − 5.94i)12-s + (1.72 + 2.99i)13-s + (−11.7 + 6.77i)14-s + (−2.00 + 3.46i)16-s + 30.5i·17-s + (3.50 − 12.2i)18-s − 10.0·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.788 − 0.615i)3-s + (0.249 + 0.433i)4-s + (0.265 + 0.655i)6-s + (0.684 − 1.18i)7-s − 0.353i·8-s + (0.242 + 0.970i)9-s + (0.803 + 0.463i)11-s + (0.0694 − 0.495i)12-s + (0.132 + 0.230i)13-s + (−0.838 + 0.484i)14-s + (−0.125 + 0.216i)16-s + 1.79i·17-s + (0.194 − 0.679i)18-s − 0.530·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ 0.559 + 0.828i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.105517353\)
\(L(\frac12)\) \(\approx\) \(1.105517353\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (2.36 + 1.84i)T \)
5 \( 1 \)
good7 \( 1 + (-4.79 + 8.30i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.83 - 5.10i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.72 - 2.99i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 30.5iT - 289T^{2} \)
19 \( 1 + 10.0T + 361T^{2} \)
23 \( 1 + (-26.5 + 15.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-36.3 - 21.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-5.01 - 8.69i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 21.9T + 1.36e3T^{2} \)
41 \( 1 + (-34.3 + 19.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-10.8 + 18.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (10.8 + 6.28i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 47.0iT - 2.80e3T^{2} \)
59 \( 1 + (-28.0 + 16.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-57.1 + 98.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.1 + 27.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 65.9iT - 5.04e3T^{2} \)
73 \( 1 + 54.1T + 5.32e3T^{2} \)
79 \( 1 + (-44.0 + 76.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-35.5 - 20.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 38.5iT - 7.92e3T^{2} \)
97 \( 1 + (64.1 - 111. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.66656048765304520644927302473, −10.29646400463692480410381382399, −8.795936940883402011469897535544, −8.011258066849484620462850000846, −6.97238759976344070114464970359, −6.44409499498247858614756091795, −4.84904589941211565962690105008, −3.88237658415755059785103253576, −1.87450757737040722224716677747, −0.935283626498210784715680001593, 0.948493022371463065269501081604, 2.81166156402013155358523486441, 4.54696326291378693540173530774, 5.42924820671149615928468459582, 6.23155852737279008500354326818, 7.27619240079683125214842876038, 8.603840045994664954531293159538, 9.152162537595736303875520729557, 9.986146759880054094488097511215, 11.20241823762783986909986380419

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