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Properties

Label 2-273-91.59-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.108 + 0.994i$
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  + (0.465 + 0.465i)2-s + (0.866 − 0.5i)3-s − 1.56i·4-s + (−3.81 − 1.02i)5-s + (0.636 + 0.170i)6-s + (−2.61 − 0.424i)7-s + (1.66 − 1.66i)8-s + (0.499 − 0.866i)9-s + (−1.30 − 2.25i)10-s + (2.60 + 0.696i)11-s + (−0.782 − 1.35i)12-s + (0.360 − 3.58i)13-s + (−1.01 − 1.41i)14-s + (−3.81 + 1.02i)15-s − 1.58·16-s + 3.34·17-s + ⋯
L(s)  = 1  + (0.329 + 0.329i)2-s + (0.499 − 0.288i)3-s − 0.782i·4-s + (−1.70 − 0.457i)5-s + (0.259 + 0.0696i)6-s + (−0.987 − 0.160i)7-s + (0.587 − 0.587i)8-s + (0.166 − 0.288i)9-s + (−0.411 − 0.713i)10-s + (0.784 + 0.210i)11-s + (−0.226 − 0.391i)12-s + (0.0999 − 0.994i)13-s + (−0.272 − 0.378i)14-s + (−0.986 + 0.264i)15-s − 0.395·16-s + 0.811·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.108 + 0.994i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765027 - 0.853011i\)
\(L(\frac12)\) \(\approx\) \(0.765027 - 0.853011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.61 + 0.424i)T \)
13 \( 1 + (-0.360 + 3.58i)T \)
good2 \( 1 + (-0.465 - 0.465i)T + 2iT^{2} \)
5 \( 1 + (3.81 + 1.02i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.60 - 0.696i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 3.34T + 17T^{2} \)
19 \( 1 + (1.71 + 6.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.53iT - 23T^{2} \)
29 \( 1 + (1.95 - 3.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.378 - 1.41i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.48 + 1.48i)T - 37iT^{2} \)
41 \( 1 + (-0.717 - 2.67i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.41 + 5.43i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.473 + 1.76i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.90 + 6.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.420 + 0.420i)T + 59iT^{2} \)
61 \( 1 + (-4.75 - 2.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.853 - 3.18i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.629 - 2.35i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.50 - 0.670i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.72 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.485 + 0.485i)T - 83iT^{2} \)
89 \( 1 + (-5.80 - 5.80i)T + 89iT^{2} \)
97 \( 1 + (7.37 + 1.97i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.78665174194164372185067583591, −10.77991397691341768963680253417, −9.597603153144253426146058687362, −8.746749062904682252327125138267, −7.48111358501832894876317996685, −6.94476316416049628874802035912, −5.54341168990516211848976460926, −4.22685767275303748919135303284, −3.31324225317928043894718459852, −0.78618404943916540978750944296, 2.77550165498902840161771961002, 3.88295805074899637720303148478, 4.13534456877651850804091541220, 6.38779995249872436238331483611, 7.46536721294449675705796693947, 8.226077799651827379533899494573, 9.146929220759014907288464394186, 10.45539518195086711832678589255, 11.47732377507993957564333492140, 12.16224262408424680432956616456

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