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Properties

Label 2-273-91.16-c1-0-13
Degree $2$
Conductor $273$
Sign $0.157 + 0.987i$
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  + 0.0680·2-s + (0.5 − 0.866i)3-s − 1.99·4-s + (1.52 − 2.64i)5-s + (0.0340 − 0.0589i)6-s + (0.910 + 2.48i)7-s − 0.271·8-s + (−0.499 − 0.866i)9-s + (0.104 − 0.180i)10-s + (2.17 − 3.77i)11-s + (−0.997 + 1.72i)12-s + (−1.79 − 3.12i)13-s + (0.0619 + 0.169i)14-s + (−1.52 − 2.64i)15-s + 3.97·16-s − 3.52·17-s + ⋯
L(s)  = 1  + 0.0481·2-s + (0.288 − 0.499i)3-s − 0.997·4-s + (0.684 − 1.18i)5-s + (0.0138 − 0.0240i)6-s + (0.344 + 0.938i)7-s − 0.0961·8-s + (−0.166 − 0.288i)9-s + (0.0329 − 0.0569i)10-s + (0.656 − 1.13i)11-s + (−0.288 + 0.498i)12-s + (−0.499 − 0.866i)13-s + (0.0165 + 0.0451i)14-s + (−0.394 − 0.684i)15-s + 0.993·16-s − 0.855·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.157 + 0.987i$
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: \((2,\ 273,\ (\ :1/2),\ 0.157 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961977 - 0.820708i\)
\(L(\frac12)\) \(\approx\) \(0.961977 - 0.820708i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.910 - 2.48i)T \)
13 \( 1 + (1.79 + 3.12i)T \)
good2 \( 1 - 0.0680T + 2T^{2} \)
5 \( 1 + (-1.52 + 2.64i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.17 + 3.77i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.52T + 17T^{2} \)
19 \( 1 + (3.45 + 5.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 + (-4.95 - 8.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.62 - 8.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.109T + 37T^{2} \)
41 \( 1 + (-1.76 - 3.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.844 - 1.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.28 + 2.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.65 - 4.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.55T + 59T^{2} \)
61 \( 1 + (2.43 + 4.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.340 - 0.589i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.61 - 4.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.75 - 3.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.85 + 8.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.41T + 83T^{2} \)
89 \( 1 + 7.70T + 89T^{2} \)
97 \( 1 + (3.86 - 6.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.04720135330592798955600121284, −10.73239173217616411014338323451, −9.281764213235000701640139497830, −8.750496274664841777614122820132, −8.400622403723656554127949088803, −6.57268496841026107078475677407, −5.36766694206233909292968577111, −4.74353230602633923276499403818, −2.89092541731920135437331883609, −1.05294394591876147595302837625, 2.20409486821433856676644571218, 4.00659113218194987846656807985, 4.52655850755521287049561168728, 6.20202260851461358929743272963, 7.17670380847384691411935694541, 8.350071274336085852266756179404, 9.693530101940407975146315925000, 9.923385715398681462891343541470, 10.90258897635548186776441400909, 12.09441396762023951030983602390

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