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Properties

Label 2-450-45.23-c1-0-16
Degree $2$
Conductor $450$
Sign $-0.893 + 0.449i$
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  + (−0.258 + 0.965i)2-s + (0.448 − 1.67i)3-s + (−0.866 − 0.499i)4-s + (1.50 + 0.866i)6-s + (−3.34 − 0.896i)7-s + (0.707 − 0.707i)8-s + (−2.59 − 1.50i)9-s + (−1.5 + 0.866i)11-s + (−1.22 + 1.22i)12-s + (−3.34 + 0.896i)13-s + (1.73 − 3.00i)14-s + (0.500 + 0.866i)16-s + (−2.12 − 2.12i)17-s + (2.12 − 2.12i)18-s + 7i·19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.258 − 0.965i)3-s + (−0.433 − 0.249i)4-s + (0.612 + 0.353i)6-s + (−1.26 − 0.338i)7-s + (0.249 − 0.249i)8-s + (−0.866 − 0.5i)9-s + (−0.452 + 0.261i)11-s + (−0.353 + 0.353i)12-s + (−0.928 + 0.248i)13-s + (0.462 − 0.801i)14-s + (0.125 + 0.216i)16-s + (−0.514 − 0.514i)17-s + (0.499 − 0.499i)18-s + 1.60i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.893 + 0.449i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0701815 - 0.295535i\)
\(L(\frac12)\) \(\approx\) \(0.0701815 - 0.295535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 \)
good7 \( 1 + (3.34 + 0.896i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.34 - 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.12 + 2.12i)T + 17iT^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 + (1.55 + 5.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.73 - 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 + (10.5 + 6.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.34 + 5.01i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.55 + 5.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (-6.06 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.448 + 1.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (-6.12 - 6.12i)T + 73iT^{2} \)
79 \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.5 - 3.10i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 + (-8.36 - 2.24i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.35926657521264054706682559000, −9.766395652824338257560322454211, −8.683200839322883107203048624793, −7.83868452970848706708874074616, −6.88112857728902040678099816360, −6.43107317196225330440480851141, −5.19987942386074437432898161575, −3.66892946758160359052545163007, −2.27761270376454924522887663157, −0.17976486837392910438750151891, 2.59253926475763213027421666137, 3.28212061155652647303128791698, 4.55854507460634459623752892171, 5.54451729171603132099424154318, 6.86108087212338270174863464056, 8.209964770353947446947515918284, 9.099176644538601385741327085984, 9.745862208924651234142332628481, 10.40341658896922658575006315891, 11.30532688372162353468304732535

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