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Properties

Label 2-273-91.24-c1-0-17
Degree $2$
Conductor $273$
Sign $0.514 + 0.857i$
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  + (2.39 − 0.640i)2-s − i·3-s + (3.57 − 2.06i)4-s + (−1.06 − 0.286i)5-s + (−0.640 − 2.39i)6-s + (0.327 + 2.62i)7-s + (3.72 − 3.72i)8-s − 9-s − 2.74·10-s + (1.52 − 1.52i)11-s + (−2.06 − 3.57i)12-s + (−3.47 − 0.970i)13-s + (2.46 + 6.06i)14-s + (−0.286 + 1.06i)15-s + (2.39 − 4.14i)16-s + (1.59 + 2.76i)17-s + ⋯
L(s)  = 1  + (1.69 − 0.453i)2-s − 0.577i·3-s + (1.78 − 1.03i)4-s + (−0.478 − 0.128i)5-s + (−0.261 − 0.976i)6-s + (0.123 + 0.992i)7-s + (1.31 − 1.31i)8-s − 0.333·9-s − 0.866·10-s + (0.461 − 0.461i)11-s + (−0.595 − 1.03i)12-s + (−0.963 − 0.269i)13-s + (0.658 + 1.62i)14-s + (−0.0739 + 0.276i)15-s + (0.598 − 1.03i)16-s + (0.387 + 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49418 - 1.41188i\)
\(L(\frac12)\) \(\approx\) \(2.49418 - 1.41188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-0.327 - 2.62i)T \)
13 \( 1 + (3.47 + 0.970i)T \)
good2 \( 1 + (-2.39 + 0.640i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.06 + 0.286i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.52 + 1.52i)T - 11iT^{2} \)
17 \( 1 + (-1.59 - 2.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.51 - 2.51i)T - 19iT^{2} \)
23 \( 1 + (-0.620 - 0.358i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.58 - 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.625 + 2.33i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.726 - 2.71i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.01 - 1.07i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.32 - 4.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.85 + 10.6i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.54 + 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.10 - 7.87i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (8.40 + 8.40i)T + 67iT^{2} \)
71 \( 1 + (14.8 - 3.96i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.86 - 1.30i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.63 + 3.63i)T - 83iT^{2} \)
89 \( 1 + (5.24 - 1.40i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.61 - 13.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.04704750653607232822983063164, −11.42583372671799940021208189078, −10.26274671226444158239805445900, −8.740223393045496756575527365666, −7.62213457249486530594363106244, −6.28319017431293147336567233236, −5.61631509249973917752372093874, −4.43002162361878128062368528134, −3.20324216165149054399855458766, −2.00295185011606709168666660006, 2.78424970894799631327685013228, 4.15106199908767045206986205769, 4.50787984300429705506173828895, 5.78561664449285115689392697137, 7.10103684506715107790968165338, 7.53449914721917487238566095899, 9.274217374034758825195252610443, 10.47956449946343368265200177330, 11.45996459625965218524879291126, 12.14676909058047471251540692475

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