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Properties

Label 2-273-91.16-c1-0-12
Degree $2$
Conductor $273$
Sign $0.997 + 0.0653i$
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  + 1.80·2-s + (−0.5 + 0.866i)3-s + 1.27·4-s + (1.98 − 3.44i)5-s + (−0.904 + 1.56i)6-s + (1.70 + 2.02i)7-s − 1.31·8-s + (−0.499 − 0.866i)9-s + (3.59 − 6.23i)10-s + (0.143 − 0.248i)11-s + (−0.637 + 1.10i)12-s + (3.60 + 0.185i)13-s + (3.09 + 3.65i)14-s + (1.98 + 3.44i)15-s − 4.92·16-s − 4.60·17-s + ⋯
L(s)  = 1  + 1.27·2-s + (−0.288 + 0.499i)3-s + 0.637·4-s + (0.888 − 1.53i)5-s + (−0.369 + 0.639i)6-s + (0.645 + 0.763i)7-s − 0.463·8-s + (−0.166 − 0.288i)9-s + (1.13 − 1.97i)10-s + (0.0432 − 0.0748i)11-s + (−0.184 + 0.318i)12-s + (0.998 + 0.0515i)13-s + (0.826 + 0.977i)14-s + (0.513 + 0.888i)15-s − 1.23·16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.997 + 0.0653i$
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: \((2,\ 273,\ (\ :1/2),\ 0.997 + 0.0653i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36509 - 0.0773440i\)
\(L(\frac12)\) \(\approx\) \(2.36509 - 0.0773440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 2.02i)T \)
13 \( 1 + (-3.60 - 0.185i)T \)
good2 \( 1 - 1.80T + 2T^{2} \)
5 \( 1 + (-1.98 + 3.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.143 + 0.248i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 + (-3.48 - 6.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 + (-0.421 - 0.730i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.212 + 0.368i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 + (0.509 + 0.883i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.585 + 1.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.71 + 4.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.574 + 0.994i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.85T + 59T^{2} \)
61 \( 1 + (4.08 + 7.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.786 - 1.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.22 - 5.58i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.24 - 14.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.84 - 6.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 2.21T + 89T^{2} \)
97 \( 1 + (-9.52 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.16337795121123223246586655506, −11.37309142356170465367893982750, −9.951005584117845129191333104426, −8.956892741733334188899348899853, −8.371249496536439961512267171134, −6.10791563222613525271447120110, −5.62811542873831296997775528270, −4.80022804685344412750573185278, −3.85880182684157349803015361985, −1.88955969786715609960266021716, 2.16595505094513585383108778501, 3.42844158162229369912561865964, 4.72974929325499092229187912370, 6.01602610533710989783978189044, 6.57810240838499819649092579247, 7.52935365613380090037343871210, 9.130695351499250834538939830770, 10.51201410324104407017100961325, 11.11406183921752627638912855684, 11.89079507337481911761307996117

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