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Properties

Label 2-273-91.31-c1-0-16
Degree $2$
Conductor $273$
Sign $0.998 - 0.0553i$
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  + (2.52 + 0.677i)2-s + (−0.866 − 0.5i)3-s + (4.20 + 2.42i)4-s + (1.05 − 3.92i)5-s + (−1.85 − 1.85i)6-s + (−2.39 + 1.12i)7-s + (5.28 + 5.28i)8-s + (0.499 + 0.866i)9-s + (5.31 − 9.20i)10-s + (0.560 − 0.150i)11-s + (−2.42 − 4.20i)12-s + (−2.59 + 2.50i)13-s + (−6.81 + 1.21i)14-s + (−2.87 + 2.87i)15-s + (4.93 + 8.54i)16-s + (0.229 − 0.397i)17-s + ⋯
L(s)  = 1  + (1.78 + 0.479i)2-s + (−0.499 − 0.288i)3-s + (2.10 + 1.21i)4-s + (0.469 − 1.75i)5-s + (−0.755 − 0.755i)6-s + (−0.905 + 0.424i)7-s + (1.86 + 1.86i)8-s + (0.166 + 0.288i)9-s + (1.68 − 2.91i)10-s + (0.169 − 0.0453i)11-s + (−0.700 − 1.21i)12-s + (−0.720 + 0.693i)13-s + (−1.82 + 0.325i)14-s + (−0.741 + 0.741i)15-s + (1.23 + 2.13i)16-s + (0.0556 − 0.0964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.998 - 0.0553i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.998 - 0.0553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86151 + 0.0792310i\)
\(L(\frac12)\) \(\approx\) \(2.86151 + 0.0792310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 1.12i)T \)
13 \( 1 + (2.59 - 2.50i)T \)
good2 \( 1 + (-2.52 - 0.677i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.05 + 3.92i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.560 + 0.150i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.229 + 0.397i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.99 - 7.44i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0405 - 0.0233i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.08T + 29T^{2} \)
31 \( 1 + (-4.88 + 1.31i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.13 + 4.23i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.50 + 4.50i)T + 41iT^{2} \)
43 \( 1 + 0.884iT - 43T^{2} \)
47 \( 1 + (-3.18 - 0.852i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.253 + 0.438i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.577 - 2.15i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.165 + 0.0958i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.31 + 8.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.23 - 4.23i)T - 71iT^{2} \)
73 \( 1 + (1.87 + 6.99i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.92 - 5.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.67 + 9.67i)T + 83iT^{2} \)
89 \( 1 + (-18.2 - 4.88i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.63 + 5.63i)T + 97iT^{2} \)
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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

−12.23279591768274817169742824048, −11.83655250297529212801848457292, −10.10110503735385619377254590738, −8.918288606418587169144073922825, −7.66475368582052160983296701404, −6.36609368385293041548713832335, −5.74708669722993131380779490174, −4.88799217512142116887759919133, −3.90928353353663713583573609436, −2.05163459145854725949635682167, 2.61301677291142607154623785220, 3.27281655973919290443964073209, 4.56018712624721750194402702981, 5.81038969815013130535371696306, 6.63447582291003329418156954866, 7.12441383614108170691214499060, 9.801582097332910510365440656867, 10.40081133480175209938932701643, 11.08005436318576920298889022269, 11.88707658635563053875491073590

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