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Properties

Label 2-273-13.10-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.0875 + 0.996i$
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  + (2.12 − 1.22i)2-s + (−0.5 − 0.866i)3-s + (2.00 − 3.47i)4-s + 0.0682i·5-s + (−2.12 − 1.22i)6-s + (0.866 + 0.5i)7-s − 4.94i·8-s + (−0.499 + 0.866i)9-s + (0.0837 + 0.145i)10-s + (0.587 − 0.339i)11-s − 4.01·12-s + (−3.24 + 1.56i)13-s + 2.45·14-s + (0.0591 − 0.0341i)15-s + (−2.05 − 3.55i)16-s + (−0.467 + 0.809i)17-s + ⋯
L(s)  = 1  + (1.50 − 0.867i)2-s + (−0.288 − 0.499i)3-s + (1.00 − 1.73i)4-s + 0.0305i·5-s + (−0.867 − 0.500i)6-s + (0.327 + 0.188i)7-s − 1.74i·8-s + (−0.166 + 0.288i)9-s + (0.0264 + 0.0458i)10-s + (0.177 − 0.102i)11-s − 1.15·12-s + (−0.900 + 0.433i)13-s + 0.655·14-s + (0.0152 − 0.00881i)15-s + (−0.513 − 0.888i)16-s + (−0.113 + 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0875 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0875 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0875 + 0.996i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0875 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70458 - 1.86092i\)
\(L(\frac12)\) \(\approx\) \(1.70458 - 1.86092i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.24 - 1.56i)T \)
good2 \( 1 + (-2.12 + 1.22i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 0.0682iT - 5T^{2} \)
11 \( 1 + (-0.587 + 0.339i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.467 - 0.809i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.79 + 1.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.14 - 3.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.00 + 6.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 + (-0.456 + 0.263i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.44 + 2.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.894 - 1.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.22iT - 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 + (7.21 + 4.16i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.505 - 0.875i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.32 - 3.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.58 + 4.95i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.42iT - 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 7.03iT - 83T^{2} \)
89 \( 1 + (-13.6 + 7.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.4 + 8.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.81951385481692178320359230502, −11.16456935124861398838143947042, −10.25613708132464025928767665436, −8.917032496351540396135588483433, −7.37785829185524052358522067779, −6.32690128371604548026946032569, −5.27498455878677716841673046805, −4.40997781319258081597196729223, −2.96937396874208553017894898956, −1.73099028008793298113723508082, 2.87088073098854890033205704952, 4.22955794928428171370991177951, 4.93985634406199888415866762311, 5.91749019755681210311099606215, 6.96865150837892769418306103205, 7.85985099357051136475527672621, 9.208850225785296519966120023373, 10.52747485514769718194783822148, 11.47281559703250642277313276532, 12.48476373497460043512665914363

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