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Properties

Label 2-273-91.34-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.921 - 0.388i$
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 + (−2.58 − 0.581i)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 + (0.581 − 2.58i)21-s + 2.16i·23-s + ⋯
L(s)  = 1  + 0.577i·3-s − i·4-s + (−1.15 + 1.15i)5-s + (−0.975 − 0.219i)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 + (0.126 − 0.563i)21-s + 0.450i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.921 - 0.388i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0734902 + 0.363450i\)
\(L(\frac12)\) \(\approx\) \(0.0734902 + 0.363450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.581i)T \)
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

−11.96111008738360655125820926208, −11.19533528139428490834484655181, −10.42363048775392310063803251524, −9.753444593991927773910011081491, −8.653160870567070452921989276919, −6.87908868386057324851750849332, −6.82084429060559135895805656085, −5.05927466064118876277340027677, −3.96380462196344232086535265518, −2.68963071339564813570180355042, 0.26936776768805653201568045932, 2.90428594755749102083070976201, 3.92598524329500373726504983588, 5.30752744628342674365248933365, 6.71210105696702462375622295506, 7.894150504156955864366333265337, 8.317289832908387279792039431917, 9.207963209906820807422913866384, 10.80180427763815537566388976130, 11.85697950077729420604548892766

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