Properties

Label 2-450-3.2-c2-0-10
Degree $2$
Conductor $450$
Sign $0.816 + 0.577i$
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  + 1.41i·2-s − 2.00·4-s + 4·7-s − 2.82i·8-s − 16.9i·11-s − 8·13-s + 5.65i·14-s + 4.00·16-s − 12.7i·17-s − 16·19-s + 24·22-s − 16.9i·23-s − 11.3i·26-s − 8.00·28-s − 4.24i·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 0.571·7-s − 0.353i·8-s − 1.54i·11-s − 0.615·13-s + 0.404i·14-s + 0.250·16-s − 0.748i·17-s − 0.842·19-s + 1.09·22-s − 0.737i·23-s − 0.435i·26-s − 0.285·28-s − 0.146i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.369507898\)
\(L(\frac12)\) \(\approx\) \(1.369507898\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4T + 49T^{2} \)
11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 + 8T + 169T^{2} \)
17 \( 1 + 12.7iT - 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 + 16.9iT - 529T^{2} \)
29 \( 1 + 4.24iT - 841T^{2} \)
31 \( 1 - 44T + 961T^{2} \)
37 \( 1 - 34T + 1.36e3T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 - 40T + 1.84e3T^{2} \)
47 \( 1 + 84.8iT - 2.20e3T^{2} \)
53 \( 1 - 38.1iT - 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 - 50T + 3.72e3T^{2} \)
67 \( 1 + 8T + 4.48e3T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 - 16T + 5.32e3T^{2} \)
79 \( 1 + 76T + 6.24e3T^{2} \)
83 \( 1 - 118. iT - 6.88e3T^{2} \)
89 \( 1 + 12.7iT - 7.92e3T^{2} \)
97 \( 1 + 176T + 9.40e3T^{2} \)
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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

−10.80594052948724572388445384316, −9.790436656198777203937546198644, −8.660324312559166151955503080685, −8.180675030583159019148595418883, −7.06028153938759640360044116695, −6.10327590941076757315290499347, −5.15798651477903772444393608744, −4.11438724139480995497635965217, −2.63725051020435494464446810732, −0.59651829725488660624628487980, 1.52612558760377130127707855492, 2.62028100789085471413374289907, 4.23323796104088499016819604541, 4.83899316457483005950746687490, 6.23586915254361910640044429833, 7.47979486438902504691639005918, 8.260640071930376535861670118409, 9.460169139486811842471359441676, 10.06068215057095015217953094954, 10.98629889995339908300020846534

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