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Properties

Label 2-450-45.14-c2-0-30
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: $\chi_{450} (149, \cdot )$
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

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

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