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Properties

Label 2-273-91.51-c1-0-16
Degree $2$
Conductor $273$
Sign $-0.339 - 0.940i$
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  + (−1.14 − 0.661i)2-s + (−0.5 − 0.866i)3-s + (−0.124 − 0.214i)4-s + (−1.98 − 1.14i)5-s + 1.32i·6-s + (−1.14 − 2.38i)7-s + 2.97i·8-s + (−0.499 + 0.866i)9-s + (1.51 + 2.62i)10-s + (−1.08 + 0.626i)11-s + (−0.124 + 0.214i)12-s + (2.78 + 2.29i)13-s + (−0.264 + 3.49i)14-s + 2.29i·15-s + (1.72 − 2.98i)16-s + (−0.418 − 0.725i)17-s + ⋯
L(s)  = 1  + (−0.810 − 0.467i)2-s + (−0.288 − 0.499i)3-s + (−0.0620 − 0.107i)4-s + (−0.887 − 0.512i)5-s + 0.540i·6-s + (−0.433 − 0.901i)7-s + 1.05i·8-s + (−0.166 + 0.288i)9-s + (0.479 + 0.830i)10-s + (−0.327 + 0.188i)11-s + (−0.0357 + 0.0620i)12-s + (0.772 + 0.635i)13-s + (−0.0705 + 0.933i)14-s + 0.591i·15-s + (0.430 − 0.745i)16-s + (−0.101 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.339 - 0.940i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0588298 + 0.0837972i\)
\(L(\frac12)\) \(\approx\) \(0.0588298 + 0.0837972i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 + 2.38i)T \)
13 \( 1 + (-2.78 - 2.29i)T \)
good2 \( 1 + (1.14 + 0.661i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.08 - 0.626i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.418 + 0.725i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.837 + 0.483i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.11 - 7.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.78T + 29T^{2} \)
31 \( 1 + (-0.553 + 0.319i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.21 - 3.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.497iT - 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + (7.83 + 4.52i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.04 + 5.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.97 + 4.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.36 + 5.82i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 5.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (3.04 - 1.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.71 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.79iT - 83T^{2} \)
89 \( 1 + (3.47 + 2.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.57iT - 97T^{2} \)
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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

−11.39510159208870927039274859624, −10.27200452115149680163051240911, −9.443232830559837135604634837940, −8.302033998498250179722986966588, −7.63354431353404535900227555624, −6.39347408707239548792127093693, −4.99131122358178512088370708975, −3.71452016664370301925312858997, −1.62165775846887326118015183632, −0.10738818357161833465737719828, 3.10275770865853027999663625325, 4.14365328581881551860270053872, 5.80392232183289683769900131501, 6.74610369555109256201452043407, 8.022505478729082192165651760549, 8.538591751461189550410027262041, 9.629627365605990529989610259824, 10.53108279946077635377291413964, 11.46680654585553498344624583947, 12.44863998249937240389465485250

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