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Properties

Label 2-273-13.10-c1-0-9
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: $\chi_{273} (127, \cdot )$
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

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

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