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Properties

Label 2-450-25.14-c1-0-5
Degree $2$
Conductor $450$
Sign $0.135 - 0.990i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (1.53 − 1.62i)5-s + 4.63i·7-s + (−0.951 + 0.309i)8-s + (2.21 + 0.292i)10-s + (−2.05 + 1.49i)11-s + (0.0846 − 0.116i)13-s + (−3.74 + 2.72i)14-s + (−0.809 − 0.587i)16-s + (7.12 − 2.31i)17-s + (2.08 + 6.41i)19-s + (1.06 + 1.96i)20-s + (−2.41 − 0.784i)22-s + (−0.985 − 1.35i)23-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (0.688 − 0.725i)5-s + 1.75i·7-s + (−0.336 + 0.109i)8-s + (0.701 + 0.0923i)10-s + (−0.619 + 0.450i)11-s + (0.0234 − 0.0323i)13-s + (−1.00 + 0.727i)14-s + (−0.202 − 0.146i)16-s + (1.72 − 0.561i)17-s + (0.477 + 1.47i)19-s + (0.238 + 0.439i)20-s + (−0.514 − 0.167i)22-s + (−0.205 − 0.282i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.135 - 0.990i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.135 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38871 + 1.21118i\)
\(L(\frac12)\) \(\approx\) \(1.38871 + 1.21118i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (-1.53 + 1.62i)T \)
good7 \( 1 - 4.63iT - 7T^{2} \)
11 \( 1 + (2.05 - 1.49i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.0846 + 0.116i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-7.12 + 2.31i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.08 - 6.41i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.985 + 1.35i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.696 - 2.14i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.310 - 0.954i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.0523 - 0.0719i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.48 - 1.80i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.02iT - 43T^{2} \)
47 \( 1 + (10.3 + 3.36i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.72 + 1.53i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.25 + 3.08i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.0 + 8.05i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-7.27 + 2.36i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.17 + 9.76i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.15 - 1.59i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.230 - 0.710i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.95 + 0.958i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.593 - 0.431i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (8.67 + 2.81i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.70332055937529236620505990954, −10.05160445813913745870535634827, −9.475343970069053085484977337852, −8.436708717195583954907786099658, −7.80937988431512849754413282575, −6.31316140198492299609511585986, −5.41338039704842812080629534529, −5.12486825631232821564916190582, −3.30355557220004678368381654799, −1.97826218316051813504853437322, 1.13447290515172601128347173120, 2.86346612343597002604544670725, 3.75851879812666824775115705693, 5.03429241760859377414764469050, 6.10775955326705444276923797246, 7.14901387380062831134566395887, 7.960333839934981191143923473097, 9.616644720146324576069985415551, 10.10799550138618568746306627391, 10.89465985500671741914525367425

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