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Properties

Label 2-273-91.16-c1-0-7
Degree $2$
Conductor $273$
Sign $0.999 - 0.00344i$
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  − 0.261·2-s + (−0.5 + 0.866i)3-s − 1.93·4-s + (0.708 − 1.22i)5-s + (0.130 − 0.226i)6-s + (2.55 − 0.693i)7-s + 1.02·8-s + (−0.499 − 0.866i)9-s + (−0.185 + 0.320i)10-s + (0.422 − 0.731i)11-s + (0.965 − 1.67i)12-s + (1.58 + 3.24i)13-s + (−0.666 + 0.181i)14-s + (0.708 + 1.22i)15-s + 3.59·16-s + 5.09·17-s + ⋯
L(s)  = 1  − 0.184·2-s + (−0.288 + 0.499i)3-s − 0.965·4-s + (0.316 − 0.548i)5-s + (0.0533 − 0.0923i)6-s + (0.964 − 0.262i)7-s + 0.363·8-s + (−0.166 − 0.288i)9-s + (−0.0585 + 0.101i)10-s + (0.127 − 0.220i)11-s + (0.278 − 0.482i)12-s + (0.438 + 0.898i)13-s + (−0.178 + 0.0484i)14-s + (0.182 + 0.316i)15-s + 0.898·16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.00344i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 - 0.00344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03968 + 0.00178879i\)
\(L(\frac12)\) \(\approx\) \(1.03968 + 0.00178879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 + 0.693i)T \)
13 \( 1 + (-1.58 - 3.24i)T \)
good2 \( 1 + 0.261T + 2T^{2} \)
5 \( 1 + (-0.708 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.422 + 0.731i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.09T + 17T^{2} \)
19 \( 1 + (1.73 + 2.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.20T + 23T^{2} \)
29 \( 1 + (4.02 + 6.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.19 - 3.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.39T + 37T^{2} \)
41 \( 1 + (5.16 + 8.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.33 - 9.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.80 + 3.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.08 + 1.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + (1.66 + 2.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.75 + 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.45 - 11.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0588 + 0.101i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.76 - 9.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.17T + 83T^{2} \)
89 \( 1 + 6.22T + 89T^{2} \)
97 \( 1 + (0.165 - 0.285i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.76025505587167956319298134624, −10.91236655885518533698930820131, −9.908509867628547696985687856424, −8.968663956310009136918103293920, −8.440635455437453142103246360969, −7.05248627329714901972781692492, −5.40182020092036420317539222630, −4.84741362711991842090771862218, −3.69848389999187630518898843823, −1.22971684329218520269418000251, 1.36792448664757590046881582080, 3.29702374559874767730556573416, 4.95418156964775801106942067623, 5.68950787724617818685495918975, 7.11580996098646495517377780914, 8.117226961347800595822769377439, 8.859272661259159835330468106725, 10.19055544226216361539533787321, 10.78118463278248140613251162369, 11.99802458471691369257474968118

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