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Properties

Label 2-450-25.19-c1-0-1
Degree $2$
Conductor $450$
Sign $0.101 - 0.994i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (2.22 − 0.166i)5-s + 5.07i·7-s + (−0.587 + 0.809i)8-s + (−2.06 + 0.847i)10-s + (0.361 + 1.11i)11-s + (−3.27 − 1.06i)13-s + (−1.56 − 4.82i)14-s + (0.309 − 0.951i)16-s + (−1.86 + 2.56i)17-s + (0.857 + 0.623i)19-s + (1.70 − 1.44i)20-s + (−0.687 − 0.945i)22-s + (0.484 − 0.157i)23-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.404 − 0.293i)4-s + (0.997 − 0.0746i)5-s + 1.91i·7-s + (−0.207 + 0.286i)8-s + (−0.654 + 0.268i)10-s + (0.108 + 0.335i)11-s + (−0.908 − 0.295i)13-s + (−0.419 − 1.29i)14-s + (0.0772 − 0.237i)16-s + (−0.452 + 0.623i)17-s + (0.196 + 0.143i)19-s + (0.381 − 0.323i)20-s + (−0.146 − 0.201i)22-s + (0.101 − 0.0328i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.101 - 0.994i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817950 + 0.738622i\)
\(L(\frac12)\) \(\approx\) \(0.817950 + 0.738622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 \)
5 \( 1 + (-2.22 + 0.166i)T \)
good7 \( 1 - 5.07iT - 7T^{2} \)
11 \( 1 + (-0.361 - 1.11i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.27 + 1.06i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.86 - 2.56i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.857 - 0.623i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.484 + 0.157i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.22 - 1.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.09 - 1.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.41 - 0.785i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.58 + 7.96i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.18iT - 43T^{2} \)
47 \( 1 + (-2.52 - 3.47i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.96 - 4.07i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.45 - 4.47i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.486 - 1.49i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.29 + 4.52i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-10.1 + 7.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-10.1 + 3.30i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.2 + 7.47i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.88 + 2.59i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.61 + 11.1i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.30 + 7.29i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.17511745161136074852525142503, −10.14379739681332053237014255274, −9.320748291762798873461015603081, −8.872072049911264636676942914496, −7.81113528797867205270586617187, −6.51868860536823663492115629353, −5.76876041914217318326248811500, −4.96146235561373259170084137753, −2.74194146982938034362459465295, −1.90607243239857289056459172843, 0.869599536643534136925069399850, 2.39984390288848025436439012250, 3.85968454534324081402546219843, 5.07620497485100582634757179740, 6.58721689965614827416514842666, 7.13946990219557637082156212947, 8.127361682584334958543868406055, 9.484490041670778293517101487389, 9.840777999069678283650961929956, 10.76737243493611046583636957105

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