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Properties

Label 2-273-273.128-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.656 - 0.754i$
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  + (−0.866 + 1.5i)3-s + 2i·4-s + (2 + 1.73i)7-s + (−1.5 − 2.59i)9-s + (−3 − 1.73i)12-s + (−3.46 + i)13-s − 4·16-s + (−1.40 + 5.23i)19-s + (−4.33 + 1.50i)21-s + (4.33 − 2.5i)25-s + 5.19·27-s + (−3.46 + 4i)28-s + (0.303 − 1.13i)31-s + (5.19 − 3i)36-s + (2.90 − 2.90i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + i·4-s + (0.755 + 0.654i)7-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (−0.960 + 0.277i)13-s − 16-s + (−0.321 + 1.20i)19-s + (−0.944 + 0.327i)21-s + (0.866 − 0.5i)25-s + 1.00·27-s + (−0.654 + 0.755i)28-s + (0.0545 − 0.203i)31-s + (0.866 − 0.5i)36-s + (0.477 − 0.477i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.656 - 0.754i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.414442 + 0.910511i\)
\(L(\frac12)\) \(\approx\) \(0.414442 + 0.910511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 1.5i)T \)
7 \( 1 + (-2 - 1.73i)T \)
13 \( 1 + (3.46 - i)T \)
good2 \( 1 - 2iT^{2} \)
5 \( 1 + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (1.40 - 5.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.303 + 1.13i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.90 + 2.90i)T - 37iT^{2} \)
41 \( 1 + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9 - 5.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (-7.79 - 13.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-15.7 + 4.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (16.2 + 4.36i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.06 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (-9.42 + 2.52i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.10053510810360830998394135880, −11.45254172024237941008814313867, −10.44723413079805183576835408851, −9.329366181684506012552081692723, −8.503278055495290341917227141772, −7.50556547156951220467652837360, −6.13225280328268682028443378643, −4.92659139781207475992090167654, −4.03740561699718515129904801903, −2.57887738356516318059680039566, 0.837384958111308657855300705166, 2.31566106793749760747816507151, 4.68146714230134789191894587126, 5.40080032976473768980367838611, 6.70332313112167076894014995047, 7.36946305867833392682029170988, 8.584732224310858673690545570098, 9.852928553944860468148739461504, 10.87987159718223555033314937332, 11.31710557776129437270344340915

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