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Properties

Label 2-273-273.230-c1-0-8
Degree $2$
Conductor $273$
Sign $0.989 + 0.145i$
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  + (−2.30 − 1.33i)2-s + (0.643 + 1.60i)3-s + (2.53 + 4.39i)4-s + 3.38·5-s + (0.655 − 4.56i)6-s + (−0.947 − 2.47i)7-s − 8.19i·8-s + (−2.17 + 2.07i)9-s + (−7.79 − 4.50i)10-s + (2.78 + 1.60i)11-s + (−5.43 + 6.91i)12-s + (3.28 − 1.47i)13-s + (−1.10 + 6.95i)14-s + (2.17 + 5.44i)15-s + (−5.82 + 10.0i)16-s + (−0.544 − 0.942i)17-s + ⋯
L(s)  = 1  + (−1.62 − 0.940i)2-s + (0.371 + 0.928i)3-s + (1.26 + 2.19i)4-s + 1.51·5-s + (0.267 − 1.86i)6-s + (−0.357 − 0.933i)7-s − 2.89i·8-s + (−0.723 + 0.690i)9-s + (−2.46 − 1.42i)10-s + (0.840 + 0.485i)11-s + (−1.56 + 1.99i)12-s + (0.912 − 0.409i)13-s + (−0.295 + 1.85i)14-s + (0.562 + 1.40i)15-s + (−1.45 + 2.52i)16-s + (−0.132 − 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.989 + 0.145i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862954 - 0.0631829i\)
\(L(\frac12)\) \(\approx\) \(0.862954 - 0.0631829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.643 - 1.60i)T \)
7 \( 1 + (0.947 + 2.47i)T \)
13 \( 1 + (-3.28 + 1.47i)T \)
good2 \( 1 + (2.30 + 1.33i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.38T + 5T^{2} \)
11 \( 1 + (-2.78 - 1.60i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.544 + 0.942i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 - 1.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.75 - 1.01i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.885 - 0.511i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.28iT - 31T^{2} \)
37 \( 1 + (-1.22 + 2.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.93 + 3.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.01 - 5.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 - 2.52iT - 53T^{2} \)
59 \( 1 + (3.46 + 6.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.252 + 0.145i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.81 - 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.91 - 3.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.6iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (-4.57 + 7.91i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.50 + 4.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.25063856123331667022173701244, −10.50757077969637670881528501336, −10.00649116841808619568757820252, −9.263199079689009968750304457336, −8.653724174812473532392321079545, −7.32243897181440889062298414969, −6.10013640227819749248641718042, −4.10318754944052391395949543673, −2.87788501082924982513510101494, −1.50267979738754851088950788784, 1.36056343929659845264067218089, 2.43509720329343602040108574972, 5.81821169192525681997865838810, 6.18566629254695862767730878165, 6.91942171558016830201814519965, 8.373926708265736721965509493214, 8.994897162929673392148767781007, 9.418423611915234031419171104648, 10.63395314865852629530900927585, 11.68564389988552054870216808018

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