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Properties

Label 2-273-13.4-c1-0-3
Degree $2$
Conductor $273$
Sign $0.983 + 0.181i$
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  + (−1.44 − 0.835i)2-s + (0.5 − 0.866i)3-s + (0.396 + 0.686i)4-s + 2.68i·5-s + (−1.44 + 0.835i)6-s + (0.866 − 0.5i)7-s + 2.01i·8-s + (−0.499 − 0.866i)9-s + (2.24 − 3.88i)10-s + (5.01 + 2.89i)11-s + 0.792·12-s + (−2.37 + 2.71i)13-s − 1.67·14-s + (2.32 + 1.34i)15-s + (2.47 − 4.29i)16-s + (0.868 + 1.50i)17-s + ⋯
L(s)  = 1  + (−1.02 − 0.590i)2-s + (0.288 − 0.499i)3-s + (0.198 + 0.343i)4-s + 1.20i·5-s + (−0.590 + 0.341i)6-s + (0.327 − 0.188i)7-s + 0.713i·8-s + (−0.166 − 0.288i)9-s + (0.709 − 1.22i)10-s + (1.51 + 0.873i)11-s + 0.228·12-s + (−0.658 + 0.752i)13-s − 0.446·14-s + (0.600 + 0.346i)15-s + (0.619 − 1.07i)16-s + (0.210 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.983 + 0.181i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.983 + 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849768 - 0.0779228i\)
\(L(\frac12)\) \(\approx\) \(0.849768 - 0.0779228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (2.37 - 2.71i)T \)
good2 \( 1 + (1.44 + 0.835i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 2.68iT - 5T^{2} \)
11 \( 1 + (-5.01 - 2.89i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.868 - 1.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.43 + 0.827i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.17 + 2.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.860 - 1.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.12iT - 31T^{2} \)
37 \( 1 + (-7.42 - 4.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.382 - 0.220i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.56 - 9.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.16iT - 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (4.85 - 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.22 + 5.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.58 - 2.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.89 - 3.40i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.600iT - 73T^{2} \)
79 \( 1 - 6.10T + 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (1.70 + 0.986i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.32 - 1.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.51805651603921166709415515371, −11.01663205858728707501466484490, −9.718961811622880212269479797703, −9.392509458966808426098204891845, −8.031369417177597284847509568821, −7.17324490838033843272156693894, −6.29248870116333158401554470088, −4.37668418539637320054340832863, −2.72697729618634814042597632130, −1.56212433082419475863969869174, 1.04955694651707827952481344434, 3.51563978339984504115143760652, 4.79611201919703252667749008055, 5.98327024334801821017320226596, 7.41659227186224006310710324170, 8.307090917571023024279023421687, 9.098386053425621588328825395061, 9.447986436124865609755202972100, 10.71579544898623463180064017641, 11.95490774246749602452563968964

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