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Properties

Label 2-450-45.29-c2-0-14
Degree $2$
Conductor $450$
Sign $-0.455 - 0.890i$
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  + (−0.707 + 1.22i)2-s + (1.73 + 2.44i)3-s + (−0.999 − 1.73i)4-s + (−4.22 + 0.389i)6-s + (5.49 + 3.17i)7-s + 2.82·8-s + (−2.99 + 8.48i)9-s + (8.17 + 4.71i)11-s + (2.51 − 5.44i)12-s + (17.0 − 9.84i)13-s + (−7.77 + 4.48i)14-s + (−2.00 + 3.46i)16-s − 1.90·17-s + (−8.27 − 9.67i)18-s − 4.69·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 + 0.816i)3-s + (−0.249 − 0.433i)4-s + (−0.704 + 0.0648i)6-s + (0.785 + 0.453i)7-s + 0.353·8-s + (−0.333 + 0.942i)9-s + (0.743 + 0.429i)11-s + (0.209 − 0.454i)12-s + (1.31 − 0.757i)13-s + (−0.555 + 0.320i)14-s + (−0.125 + 0.216i)16-s − 0.112·17-s + (−0.459 − 0.537i)18-s − 0.247·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.455 - 0.890i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ -0.455 - 0.890i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.949104823\)
\(L(\frac12)\) \(\approx\) \(1.949104823\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (-1.73 - 2.44i)T \)
5 \( 1 \)
good7 \( 1 + (-5.49 - 3.17i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.17 - 4.71i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-17.0 + 9.84i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 1.90T + 289T^{2} \)
19 \( 1 + 4.69T + 361T^{2} \)
23 \( 1 + (-4.71 - 8.17i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.84 - 1.64i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-20.5 - 35.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 17.3iT - 1.36e3T^{2} \)
41 \( 1 + (53.5 - 30.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.826 + 0.477i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (7.05 - 12.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 9.53T + 2.80e3T^{2} \)
59 \( 1 + (79.2 - 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (26.8 - 15.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 85.9iT - 5.04e3T^{2} \)
73 \( 1 + 96.0iT - 5.32e3T^{2} \)
79 \( 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-43.9 + 76.1i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (-83.0 - 47.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.91334475152372471095044841043, −10.17223530378170291030135733209, −9.113340236426571035165074806535, −8.535282628700401403624324845498, −7.82810249548643451793406333514, −6.51990509901788397658162562921, −5.40974229605564068471789147542, −4.51003586083080936321769476797, −3.30135069558009102085854588276, −1.60478887184860539062542063932, 0.979639361914583887610375072374, 1.94646285531932939626986162584, 3.42765035755847002282668295940, 4.36662221632345458018163760801, 6.13561256319330624658081089847, 7.00194237460557191130842139193, 8.161899609130393202867737996137, 8.637596047120166940566569421989, 9.529991064920175944824321345066, 10.79648917918126357509827762075

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