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Properties

Label 2-273-91.4-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.599 + 0.800i$
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  − 2.04i·2-s + (0.5 + 0.866i)3-s − 2.19·4-s + (0.341 − 0.197i)5-s + (1.77 − 1.02i)6-s + (−0.908 − 2.48i)7-s + 0.400i·8-s + (−0.499 + 0.866i)9-s + (−0.403 − 0.699i)10-s + (1.35 − 0.783i)11-s + (−1.09 − 1.90i)12-s + (1.46 − 3.29i)13-s + (−5.09 + 1.85i)14-s + (0.341 + 0.197i)15-s − 3.57·16-s + 6.82·17-s + ⋯
L(s)  = 1  − 1.44i·2-s + (0.288 + 0.499i)3-s − 1.09·4-s + (0.152 − 0.0881i)5-s + (0.724 − 0.418i)6-s + (−0.343 − 0.939i)7-s + 0.141i·8-s + (−0.166 + 0.288i)9-s + (−0.127 − 0.221i)10-s + (0.409 − 0.236i)11-s + (−0.316 − 0.548i)12-s + (0.406 − 0.913i)13-s + (−1.36 + 0.497i)14-s + (0.0881 + 0.0508i)15-s − 0.892·16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606604 - 1.21184i\)
\(L(\frac12)\) \(\approx\) \(0.606604 - 1.21184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.908 + 2.48i)T \)
13 \( 1 + (-1.46 + 3.29i)T \)
good2 \( 1 + 2.04iT - 2T^{2} \)
5 \( 1 + (-0.341 + 0.197i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.35 + 0.783i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + (6.75 + 3.89i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.78T + 23T^{2} \)
29 \( 1 + (3.94 - 6.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.14 - 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.17iT - 37T^{2} \)
41 \( 1 + (-6.17 - 3.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.65 - 6.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.92 - 2.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.964 - 1.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.4iT - 59T^{2} \)
61 \( 1 + (-6.13 + 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 1.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.66 + 4.42i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.23 + 1.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.55 - 2.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.01iT - 83T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 + (9.58 - 5.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.15363212645074883885747412798, −10.80839070024480206254287044620, −9.873242675951120617807525123468, −9.188574875134445139296070819501, −7.952280954066034001258733881012, −6.57140091314802465184625409584, −4.97012067060083507566686522120, −3.72266623472418909678653539325, −3.01660916203946791891602468380, −1.13547271379064960349801822278, 2.24471897511092605360925931817, 4.12254793509886734783143261593, 5.76532083052013512741321210012, 6.21650195373645936313095673631, 7.26433115461013202933083212271, 8.254934142422238821871613712769, 8.947577495215366687506925193169, 9.965935469051083085491708485024, 11.57045512156269993782149530633, 12.36297630529748153883321251424

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