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Properties

Label 2-273-273.257-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.842 + 0.538i$
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  − 1.59·2-s + (−0.201 + 1.72i)3-s + 0.554·4-s + (−1.21 − 0.702i)5-s + (0.321 − 2.74i)6-s + (0.626 + 2.57i)7-s + 2.31·8-s + (−2.91 − 0.691i)9-s + (1.94 + 1.12i)10-s + (−0.817 + 1.41i)11-s + (−0.111 + 0.953i)12-s + (−3.59 + 0.224i)13-s + (−1.00 − 4.10i)14-s + (1.45 − 1.95i)15-s − 4.80·16-s − 0.339·17-s + ⋯
L(s)  = 1  − 1.13·2-s + (−0.116 + 0.993i)3-s + 0.277·4-s + (−0.543 − 0.314i)5-s + (0.131 − 1.12i)6-s + (0.236 + 0.971i)7-s + 0.816·8-s + (−0.973 − 0.230i)9-s + (0.614 + 0.354i)10-s + (−0.246 + 0.426i)11-s + (−0.0321 + 0.275i)12-s + (−0.998 + 0.0623i)13-s + (−0.267 − 1.09i)14-s + (0.375 − 0.503i)15-s − 1.20·16-s − 0.0823·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.842 + 0.538i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.842 + 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0295348 - 0.100970i\)
\(L(\frac12)\) \(\approx\) \(0.0295348 - 0.100970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.201 - 1.72i)T \)
7 \( 1 + (-0.626 - 2.57i)T \)
13 \( 1 + (3.59 - 0.224i)T \)
good2 \( 1 + 1.59T + 2T^{2} \)
5 \( 1 + (1.21 + 0.702i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.817 - 1.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.339T + 17T^{2} \)
19 \( 1 + (2.35 + 4.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.57iT - 23T^{2} \)
29 \( 1 + (1.65 - 0.955i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.995 + 1.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + (7.89 - 4.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 - 3.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.97 + 2.87i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.54 + 4.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.64iT - 59T^{2} \)
61 \( 1 + (-0.243 + 0.140i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.91 - 5.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.38 - 12.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.42 + 2.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.52 + 4.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.193iT - 83T^{2} \)
89 \( 1 - 4.97iT - 89T^{2} \)
97 \( 1 + (-1.17 + 2.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.05580872634739398081427707859, −11.34832355839184963900786689037, −10.23370374704375823825092484463, −9.646034551355167746970197819359, −8.599644383384293388096972621556, −8.228883940269261961745258354849, −6.74680655639770142013960024344, −5.05791262071212082753364750860, −4.44727871944581476091847411394, −2.50477151580027065086547394851, 0.11530234261953637450346777469, 1.77903430500144117542053705868, 3.73254392308310933914403561043, 5.36694723967422105001758897931, 6.97619139863224335092570797155, 7.57347448621783145408147904648, 8.163057608841666361242760577250, 9.356010817413395988616458433646, 10.47235287717353171637906601627, 11.13247815610688353072273003281

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