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Properties

Label 2-273-91.20-c1-0-16
Degree $2$
Conductor $273$
Sign $0.939 + 0.343i$
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 + (−2.62 + 0.304i)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 + (−0.993 + 0.114i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.939 + 0.343i$
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: \((2,\ 273,\ (\ :1/2),\ 0.939 + 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05889 - 0.541390i\)
\(L(\frac12)\) \(\approx\) \(3.05889 - 0.541390i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.304i)T \)
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

−12.06512741704244505523715788950, −11.24617542421438358334177340622, −10.10410555173069112274442587835, −9.344087614624883732373556004290, −7.61730982893076719912138180238, −6.63686145778906491274654438096, −5.46819744623722236313932415877, −4.42937477811383185048876077072, −3.29483985174119828438743479533, −2.52560681420254994670532533545, 2.53267095718988032094512982323, 3.65450930011782570695541105536, 4.60611979614033282983409495369, 5.95186716235520219883513406800, 6.77033854220346058450316614511, 7.63731994676902628158709251756, 8.934349748407110664510516034885, 10.14201129919872972601137526455, 11.54791146648497143609600720884, 12.45245088661856905790421230326

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