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Properties

Label 2-273-273.44-c1-0-12
Degree $2$
Conductor $273$
Sign $0.590 - 0.807i$
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  + (1.27 − 0.341i)2-s + (−0.898 + 1.48i)3-s + (−0.226 + 0.130i)4-s + (3.02 − 0.810i)5-s + (−0.638 + 2.19i)6-s + (−0.631 + 2.56i)7-s + (−2.10 + 2.10i)8-s + (−1.38 − 2.66i)9-s + (3.57 − 2.06i)10-s + (5.55 + 1.48i)11-s + (0.00980 − 0.453i)12-s + (−3.53 − 0.724i)13-s + (0.0722 + 3.48i)14-s + (−1.51 + 5.21i)15-s + (−1.70 + 2.95i)16-s + (1.21 + 2.10i)17-s + ⋯
L(s)  = 1  + (0.900 − 0.241i)2-s + (−0.518 + 0.855i)3-s + (−0.113 + 0.0654i)4-s + (1.35 − 0.362i)5-s + (−0.260 + 0.895i)6-s + (−0.238 + 0.971i)7-s + (−0.745 + 0.745i)8-s + (−0.462 − 0.886i)9-s + (1.13 − 0.653i)10-s + (1.67 + 0.448i)11-s + (0.00283 − 0.130i)12-s + (−0.979 − 0.200i)13-s + (0.0193 + 0.932i)14-s + (−0.391 + 1.34i)15-s + (−0.425 + 0.737i)16-s + (0.295 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.590 - 0.807i$
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: \((2,\ 273,\ (\ :1/2),\ 0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59993 + 0.812218i\)
\(L(\frac12)\) \(\approx\) \(1.59993 + 0.812218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.898 - 1.48i)T \)
7 \( 1 + (0.631 - 2.56i)T \)
13 \( 1 + (3.53 + 0.724i)T \)
good2 \( 1 + (-1.27 + 0.341i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-3.02 + 0.810i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-5.55 - 1.48i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.21 - 2.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.330 - 1.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.91 + 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.05iT - 29T^{2} \)
31 \( 1 + (5.49 + 1.47i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.73 + 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.64 + 3.64i)T + 41iT^{2} \)
43 \( 1 + 3.28iT - 43T^{2} \)
47 \( 1 + (0.259 + 0.969i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.84 + 1.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.87 + 1.57i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.884 + 1.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 3.04i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.00 + 3.00i)T + 71iT^{2} \)
73 \( 1 + (1.23 - 4.59i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.60 - 7.60i)T + 83iT^{2} \)
89 \( 1 + (0.620 + 2.31i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.02 - 9.02i)T - 97iT^{2} \)
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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

−12.29822371036500924964640438529, −11.38812533010003228530360812277, −9.940013951778146677115406469542, −9.391270211322138596176106082210, −8.691289945434914164453786326025, −6.44501843135476597158637842145, −5.71847453336405204461508104472, −4.93086979421269682184644533096, −3.83640561905726802510976648044, −2.32331060340219827364286794686, 1.33317053062825010656982856663, 3.24648154250248003661986823256, 4.81345903881389123815454696078, 5.78582278628046952824813095162, 6.65973020257450507662644371079, 7.18637470552169339589658406443, 9.165496377218428430579407972509, 9.794352791143445550592884344901, 11.00937243237784809220532375587, 12.01009270015408654639212701982

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