Properties

Label 2-273-91.89-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.530 - 0.847i$
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  + (1.93 + 1.93i)2-s + (−0.866 − 0.5i)3-s + 5.47i·4-s + (−0.212 − 0.792i)5-s + (−0.707 − 2.64i)6-s + (2.21 + 1.44i)7-s + (−6.71 + 6.71i)8-s + (0.499 + 0.866i)9-s + (1.12 − 1.94i)10-s + (0.207 + 0.774i)11-s + (2.73 − 4.74i)12-s + (−2.33 − 2.75i)13-s + (1.48 + 7.07i)14-s + (−0.212 + 0.792i)15-s − 15.0·16-s + 0.290·17-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)2-s + (−0.499 − 0.288i)3-s + 2.73i·4-s + (−0.0949 − 0.354i)5-s + (−0.288 − 1.07i)6-s + (0.837 + 0.546i)7-s + (−2.37 + 2.37i)8-s + (0.166 + 0.288i)9-s + (0.354 − 0.613i)10-s + (0.0625 + 0.233i)11-s + (0.790 − 1.36i)12-s + (−0.646 − 0.763i)13-s + (0.397 + 1.89i)14-s + (−0.0547 + 0.204i)15-s − 3.75·16-s + 0.0704·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.530 - 0.847i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05091 + 1.89847i\)
\(L(\frac12)\) \(\approx\) \(1.05091 + 1.89847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.21 - 1.44i)T \)
13 \( 1 + (2.33 + 2.75i)T \)
good2 \( 1 + (-1.93 - 1.93i)T + 2iT^{2} \)
5 \( 1 + (0.212 + 0.792i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.207 - 0.774i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.290T + 17T^{2} \)
19 \( 1 + (-2.05 - 0.551i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 9.32iT - 23T^{2} \)
29 \( 1 + (-2.87 - 4.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.03 - 1.35i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.61 + 1.61i)T - 37iT^{2} \)
41 \( 1 + (8.80 + 2.35i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.71 - 1.56i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.70 + 1.25i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.69 + 4.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.80 + 8.80i)T + 59iT^{2} \)
61 \( 1 + (5.06 - 2.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.2 - 3.28i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.43 - 1.72i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.07 - 7.74i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.75 + 4.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.51 + 3.51i)T - 83iT^{2} \)
89 \( 1 + (3.11 + 3.11i)T + 89iT^{2} \)
97 \( 1 + (-3.20 - 11.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.20952273825085114534822350373, −12.10210753283496935278043570738, −10.62381591639824784654117068904, −8.727778883955717961057140974448, −8.045841028901945662585632394321, −7.07066606084993475037462868940, −6.10792599951346794890735343023, −5.04606814739562419838757298676, −4.60170376308999975146838534340, −2.79518942555544649793353302012, 1.43972646139327352359940333219, 3.06573015213268561867225010150, 4.26763359258102722389352108453, 4.99914706286266239839644636074, 6.08601416053373501783526522898, 7.34498428208257514954468593990, 9.314569799727879973303837772117, 10.17665365434840644032552192924, 10.99660447879608626217582439490, 11.66928502898068989534609779134

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