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Properties

Label 2-273-91.20-c1-0-17
Degree $2$
Conductor $273$
Sign $0.449 + 0.893i$
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  + (2.37 − 0.636i)2-s + (−0.866 − 0.5i)3-s + (3.50 − 2.02i)4-s + (0.498 − 0.498i)5-s + (−2.37 − 0.636i)6-s + (−2.12 − 1.57i)7-s + (3.56 − 3.56i)8-s + (0.499 + 0.866i)9-s + (0.866 − 1.50i)10-s + (0.184 + 0.688i)11-s − 4.05·12-s + (3.17 + 1.70i)13-s + (−6.05 − 2.39i)14-s + (−0.680 + 0.182i)15-s + (2.15 − 3.73i)16-s + (2.27 + 3.93i)17-s + ⋯
L(s)  = 1  + (1.68 − 0.450i)2-s + (−0.499 − 0.288i)3-s + (1.75 − 1.01i)4-s + (0.222 − 0.222i)5-s + (−0.970 − 0.259i)6-s + (−0.802 − 0.596i)7-s + (1.26 − 1.26i)8-s + (0.166 + 0.288i)9-s + (0.274 − 0.474i)10-s + (0.0556 + 0.207i)11-s − 1.16·12-s + (0.881 + 0.471i)13-s + (−1.61 − 0.640i)14-s + (−0.175 + 0.0471i)15-s + (0.538 − 0.932i)16-s + (0.551 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.449 + 0.893i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23854 - 1.37930i\)
\(L(\frac12)\) \(\approx\) \(2.23854 - 1.37930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 1.57i)T \)
13 \( 1 + (-3.17 - 1.70i)T \)
good2 \( 1 + (-2.37 + 0.636i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.498 + 0.498i)T - 5iT^{2} \)
11 \( 1 + (-0.184 - 0.688i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.27 - 3.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.24 + 0.870i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.67 + 0.964i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.185 + 0.322i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.53 - 3.53i)T - 31iT^{2} \)
37 \( 1 + (0.545 + 2.03i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.11 + 11.6i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.38 + 3.68i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.55 - 3.55i)T + 47iT^{2} \)
53 \( 1 + 4.97T + 53T^{2} \)
59 \( 1 + (-1.03 + 3.85i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.0 - 5.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.5 - 3.37i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.10 + 7.86i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.608 + 0.608i)T + 73iT^{2} \)
79 \( 1 - 9.81T + 79T^{2} \)
83 \( 1 + (-2.25 + 2.25i)T - 83iT^{2} \)
89 \( 1 + (-17.5 + 4.70i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (8.73 + 2.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.18747321255116713863512937683, −10.89568717568144137196078338511, −10.50115841790072618972444522854, −9.011504260294031529541765011349, −7.28048679646561186536790172601, −6.31244285270230574353291480876, −5.65260280037465770637010591096, −4.32721696508102920617729309512, −3.47260718895758310865598082188, −1.74820065451790193456297411573, 2.77149155063014620260143568789, 3.79413608204705432137866639903, 5.03671004835446243891002565074, 6.06664442088924443075333260132, 6.41179828853232141330741258283, 7.81776908804564933952907513001, 9.308336112802654524502269446417, 10.47687402809350631327851344463, 11.52006140898717477845112742770, 12.26096648537033797366875378506

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