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Properties

Label 2-273-273.44-c1-0-20
Degree $2$
Conductor $273$
Sign $-0.485 + 0.873i$
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  + (−0.623 + 0.167i)2-s + (−1.63 − 0.579i)3-s + (−1.37 + 0.791i)4-s + (2.98 − 0.801i)5-s + (1.11 + 0.0887i)6-s + (−2.60 + 0.461i)7-s + (1.63 − 1.63i)8-s + (2.32 + 1.89i)9-s + (−1.72 + 0.998i)10-s + (−1.79 − 0.479i)11-s + (2.69 − 0.497i)12-s + (−2.76 − 2.31i)13-s + (1.54 − 0.722i)14-s + (−5.34 − 0.425i)15-s + (0.837 − 1.45i)16-s + (−1.98 − 3.44i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.118i)2-s + (−0.942 − 0.334i)3-s + (−0.685 + 0.395i)4-s + (1.33 − 0.358i)5-s + (0.454 + 0.0362i)6-s + (−0.984 + 0.174i)7-s + (0.578 − 0.578i)8-s + (0.775 + 0.630i)9-s + (−0.546 + 0.315i)10-s + (−0.540 − 0.144i)11-s + (0.778 − 0.143i)12-s + (−0.765 − 0.643i)13-s + (0.413 − 0.193i)14-s + (−1.37 − 0.109i)15-s + (0.209 − 0.362i)16-s + (−0.482 − 0.835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.485 + 0.873i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.485 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198290 - 0.337131i\)
\(L(\frac12)\) \(\approx\) \(0.198290 - 0.337131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 + 0.579i)T \)
7 \( 1 + (2.60 - 0.461i)T \)
13 \( 1 + (2.76 + 2.31i)T \)
good2 \( 1 + (0.623 - 0.167i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.98 + 0.801i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.79 + 0.479i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.98 + 3.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.93 + 7.22i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.29 - 2.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.80iT - 29T^{2} \)
31 \( 1 + (0.388 + 0.104i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.52 - 2.28i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.30 - 7.30i)T + 41iT^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + (0.515 + 1.92i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.46 + 0.845i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.21 - 1.93i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.418 + 0.724i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.34 + 0.895i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.94 + 1.94i)T + 71iT^{2} \)
73 \( 1 + (-0.241 + 0.902i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.833 - 1.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + (-0.330 - 1.23i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.39 + 6.39i)T - 97iT^{2} \)
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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

−11.66420036944414270245063983297, −10.28942992227615362335346627606, −9.744891670981452299448000068015, −8.967620055802862709336993694779, −7.58353394948452081041943685337, −6.58584886644718484403232137931, −5.52592234515134073037267165120, −4.67382452923522065539788171723, −2.56620944139012970556045178720, −0.37900389394142313411787513988, 1.88076952118774979751138311678, 4.01250473956576962381019557767, 5.35824992914834667706083336667, 6.05583120594617292485223689096, 7.05052561092313274271554451697, 8.813285289533412127487644529531, 9.679517722114440136558513955667, 10.36839837564194528159120612194, 10.62231303557480759031219809813, 12.39441769006569207924817069986

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