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Properties

Label 2-273-91.83-c1-0-6
Degree $2$
Conductor $273$
Sign $0.0438 - 0.999i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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·3-s + 2i·4-s + (2.58 + 2.58i)5-s + (0.581 − 2.58i)7-s − 9-s + (−2.16 − 2.16i)11-s − 2·12-s + (−0.418 + 3.58i)13-s + (−2.58 + 2.58i)15-s − 4·16-s + 5.16·17-s + (−3.58 − 3.58i)19-s + (−5.16 + 5.16i)20-s + (2.58 + 0.581i)21-s − 2.16i·23-s + ⋯
L(s)  = 1  + 0.577i·3-s + i·4-s + (1.15 + 1.15i)5-s + (0.219 − 0.975i)7-s − 0.333·9-s + (−0.651 − 0.651i)11-s − 0.577·12-s + (−0.116 + 0.993i)13-s + (−0.666 + 0.666i)15-s − 16-s + 1.25·17-s + (−0.821 − 0.821i)19-s + (−1.15 + 1.15i)20-s + (0.563 + 0.126i)21-s − 0.450i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0438 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0438 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0438 - 0.999i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.0438 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05672 + 1.01133i\)
\(L(\frac12)\) \(\approx\) \(1.05672 + 1.01133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-0.581 + 2.58i)T \)
13 \( 1 + (0.418 - 3.58i)T \)
good2 \( 1 - 2iT^{2} \)
5 \( 1 + (-2.58 - 2.58i)T + 5iT^{2} \)
11 \( 1 + (2.16 + 2.16i)T + 11iT^{2} \)
17 \( 1 - 5.16T + 17T^{2} \)
19 \( 1 + (3.58 + 3.58i)T + 19iT^{2} \)
23 \( 1 + 2.16iT - 23T^{2} \)
29 \( 1 - 8.16T + 29T^{2} \)
31 \( 1 + (-2.41 - 2.41i)T + 31iT^{2} \)
37 \( 1 + (8.32 + 8.32i)T + 37iT^{2} \)
41 \( 1 + (-0.837 - 0.837i)T + 41iT^{2} \)
43 \( 1 - 7.32iT - 43T^{2} \)
47 \( 1 + (-3.41 + 3.41i)T - 47iT^{2} \)
53 \( 1 - 2.16T + 53T^{2} \)
59 \( 1 + (4.32 - 4.32i)T - 59iT^{2} \)
61 \( 1 + 3.16iT - 61T^{2} \)
67 \( 1 + (-6.32 + 6.32i)T - 67iT^{2} \)
71 \( 1 + (-6 + 6i)T - 71iT^{2} \)
73 \( 1 + (2.41 - 2.41i)T - 73iT^{2} \)
79 \( 1 - 1.32T + 79T^{2} \)
83 \( 1 + (-3.41 - 3.41i)T + 83iT^{2} \)
89 \( 1 + (-8.58 + 8.58i)T - 89iT^{2} \)
97 \( 1 + (-3.58 - 3.58i)T + 97iT^{2} \)
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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

−12.03519103758617799933593469388, −10.83422273714753968138557979545, −10.49138015892939372079959778931, −9.397756366775233122249200311139, −8.280920157273992439576271577395, −7.13620835045515447514980192708, −6.34328307374327436666686718148, −4.83358061923825864662554478684, −3.53601177528004863786543362319, −2.50501054311055669118991620702, 1.30392652492862420512160130643, 2.40223591361940046914155219368, 5.04337118978101835240087256500, 5.48536526890586633882438120054, 6.33448984027236089576509141357, 7.999279660337722590277705734908, 8.811081513300056959973063548806, 9.918155527056744165208627653653, 10.35419095113240899644285867131, 12.05281292180118208941371688469

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