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Properties

Label 2-273-91.76-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.845 - 0.534i$
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  + (−0.281 + 1.04i)2-s + (−0.866 + 0.5i)3-s + (0.710 + 0.410i)4-s + (−1.02 + 1.02i)5-s + (−0.281 − 1.04i)6-s + (2.23 + 1.41i)7-s + (−2.16 + 2.16i)8-s + (0.499 − 0.866i)9-s + (−0.788 − 1.36i)10-s + (0.972 + 0.260i)11-s − 0.820·12-s + (−3.05 − 1.91i)13-s + (−2.11 + 1.94i)14-s + (0.376 − 1.40i)15-s + (−0.842 − 1.45i)16-s + (−2.37 + 4.10i)17-s + ⋯
L(s)  = 1  + (−0.198 + 0.741i)2-s + (−0.499 + 0.288i)3-s + (0.355 + 0.205i)4-s + (−0.459 + 0.459i)5-s + (−0.114 − 0.428i)6-s + (0.843 + 0.536i)7-s + (−0.765 + 0.765i)8-s + (0.166 − 0.288i)9-s + (−0.249 − 0.432i)10-s + (0.293 + 0.0785i)11-s − 0.236·12-s + (−0.847 − 0.530i)13-s + (−0.565 + 0.519i)14-s + (0.0970 − 0.362i)15-s + (−0.210 − 0.364i)16-s + (−0.574 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.845 - 0.534i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272503 + 0.940819i\)
\(L(\frac12)\) \(\approx\) \(0.272503 + 0.940819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 1.41i)T \)
13 \( 1 + (3.05 + 1.91i)T \)
good2 \( 1 + (0.281 - 1.04i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.02 - 1.02i)T - 5iT^{2} \)
11 \( 1 + (-0.972 - 0.260i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.37 - 4.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.391 - 1.46i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.337 - 0.194i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.30 + 7.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.92 + 2.92i)T - 31iT^{2} \)
37 \( 1 + (-9.77 - 2.61i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.64 - 2.31i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.28 + 4.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.78 - 4.78i)T + 47iT^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + (-0.889 + 0.238i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.36 - 4.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 6.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.56 - 0.687i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.46 + 2.46i)T + 73iT^{2} \)
79 \( 1 - 7.06T + 79T^{2} \)
83 \( 1 + (-2.43 + 2.43i)T - 83iT^{2} \)
89 \( 1 + (-2.79 + 10.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.55 - 13.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.88871831099115786127621644582, −11.52002123852831999714256492463, −10.56108282231685683177475567612, −9.304134445091425973373516574647, −8.098452344670755847539817335674, −7.52761964042156651241375107231, −6.32129038783945177403563397269, −5.48786878227653781066709834446, −4.10562371460976574793981795297, −2.42549742520588268221457450664, 0.861620073255670591240264023178, 2.35062658224171728784779631024, 4.17608647495530240206943412138, 5.22256410882269443509175753918, 6.73317692344992429245524861407, 7.44972409652166124654536303403, 8.796262236041642074485169172470, 9.817790023512348930975248645003, 10.87652322598148860040638406293, 11.52224272658943402834826881497

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