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Properties

Label 2-450-5.4-c1-0-4
Degree $2$
Conductor $450$
Sign $0.894 - 0.447i$
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  + i·2-s − 4-s − 2i·7-s − i·8-s + 6·11-s − 4i·13-s + 2·14-s + 16-s + 6i·17-s + 4·19-s + 6i·22-s + 4·26-s + 2i·28-s + 6·29-s − 4·31-s + i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s + 1.80·11-s − 1.10i·13-s + 0.534·14-s + 0.250·16-s + 1.45i·17-s + 0.917·19-s + 1.27i·22-s + 0.784·26-s + 0.377i·28-s + 1.11·29-s − 0.718·31-s + 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41124 + 0.333150i\)
\(L(\frac12)\) \(\approx\) \(1.41124 + 0.333150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−11.04238032527484228369273696027, −10.16572437153849807630074510989, −9.261949268368779885007966893660, −8.327062825083179742073637693113, −7.42803712387726107542782245603, −6.52060427724931337192128274937, −5.67061075796970096100998333241, −4.30730624874092149814131848953, −3.47831764970121543239869758483, −1.19207266165402611601661331382, 1.40514541273076448597784095033, 2.82189013550091374750763219830, 4.04883168171756887383131827482, 5.08681591364366779099800787329, 6.35378265604245827247284872400, 7.27066054679861735713409817004, 8.822788090344997879108240222199, 9.172244035568603665863444619636, 10.00527370541858331902712317373, 11.41259106026623354709253200328

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