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Properties

Label 2-273-273.44-c1-0-9
Degree $2$
Conductor $273$
Sign $0.349 - 0.936i$
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  + (−1.87 + 0.501i)2-s + (1.45 + 0.933i)3-s + (1.51 − 0.874i)4-s + (−1.33 + 0.358i)5-s + (−3.19 − 1.01i)6-s + (1.90 − 1.83i)7-s + (0.343 − 0.343i)8-s + (1.25 + 2.72i)9-s + (2.32 − 1.34i)10-s + (2.28 + 0.612i)11-s + (3.02 + 0.138i)12-s + (2.20 + 2.85i)13-s + (−2.64 + 4.38i)14-s + (−2.28 − 0.726i)15-s + (−2.21 + 3.84i)16-s + (1.10 + 1.90i)17-s + ⋯
L(s)  = 1  + (−1.32 + 0.354i)2-s + (0.842 + 0.538i)3-s + (0.757 − 0.437i)4-s + (−0.598 + 0.160i)5-s + (−1.30 − 0.414i)6-s + (0.720 − 0.693i)7-s + (0.121 − 0.121i)8-s + (0.419 + 0.907i)9-s + (0.735 − 0.424i)10-s + (0.689 + 0.184i)11-s + (0.873 + 0.0398i)12-s + (0.612 + 0.790i)13-s + (−0.706 + 1.17i)14-s + (−0.590 − 0.187i)15-s + (−0.554 + 0.960i)16-s + (0.267 + 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.349 - 0.936i$
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: \((2,\ 273,\ (\ :1/2),\ 0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718575 + 0.498742i\)
\(L(\frac12)\) \(\approx\) \(0.718575 + 0.498742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.45 - 0.933i)T \)
7 \( 1 + (-1.90 + 1.83i)T \)
13 \( 1 + (-2.20 - 2.85i)T \)
good2 \( 1 + (1.87 - 0.501i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.33 - 0.358i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.28 - 0.612i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.10 - 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.11 + 4.17i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.362 - 0.628i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.61iT - 29T^{2} \)
31 \( 1 + (-5.52 - 1.48i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.96 - 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.19 - 6.19i)T + 41iT^{2} \)
43 \( 1 + 1.48iT - 43T^{2} \)
47 \( 1 + (1.11 + 4.15i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-10.3 + 5.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.06 + 1.89i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.579 - 0.155i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (10.8 + 10.8i)T + 71iT^{2} \)
73 \( 1 + (2.05 - 7.65i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.29 + 5.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.03 + 1.03i)T + 83iT^{2} \)
89 \( 1 + (2.90 + 10.8i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.431 - 0.431i)T - 97iT^{2} \)
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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

−11.60935813399133201523343323643, −10.82726545299518321858714428381, −9.997280879349516166020183804955, −9.002040135827656181890626965623, −8.400827281656631624189440625869, −7.52656298966424312169776975887, −6.73808779761856230265743692418, −4.57664986177718232157551560199, −3.71234421102101933678756435564, −1.58791455377028226864509987018, 1.14172392696500628904318678270, 2.51581723060248619702206752892, 4.08862554939117879436566661109, 5.90311318542662237910036354872, 7.45361450575930581538206626220, 8.163729583783990658757764975835, 8.659117253148627130681426367444, 9.569842289247983139844018901922, 10.61759506373068035188386892467, 11.77764805555167164518258405470

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