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Properties

Label 2-273-91.31-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.617 - 0.786i$
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  + (−0.263 − 0.0704i)2-s + (0.866 + 0.5i)3-s + (−1.66 − 0.962i)4-s + (−0.650 + 2.42i)5-s + (−0.192 − 0.192i)6-s + (−2.34 + 1.21i)7-s + (0.755 + 0.755i)8-s + (0.499 + 0.866i)9-s + (0.342 − 0.592i)10-s + (−1.62 + 0.435i)11-s + (−0.962 − 1.66i)12-s + (−3.50 − 0.834i)13-s + (0.703 − 0.154i)14-s + (−1.77 + 1.77i)15-s + (1.78 + 3.08i)16-s + (−1.60 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.185 − 0.0498i)2-s + (0.499 + 0.288i)3-s + (−0.833 − 0.481i)4-s + (−0.290 + 1.08i)5-s + (−0.0786 − 0.0786i)6-s + (−0.887 + 0.460i)7-s + (0.267 + 0.267i)8-s + (0.166 + 0.288i)9-s + (0.108 − 0.187i)10-s + (−0.489 + 0.131i)11-s + (−0.277 − 0.481i)12-s + (−0.972 − 0.231i)13-s + (0.188 − 0.0413i)14-s + (−0.458 + 0.458i)15-s + (0.445 + 0.770i)16-s + (−0.390 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.617 - 0.786i$
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: \((2,\ 273,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278044 + 0.571636i\)
\(L(\frac12)\) \(\approx\) \(0.278044 + 0.571636i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.34 - 1.21i)T \)
13 \( 1 + (3.50 + 0.834i)T \)
good2 \( 1 + (0.263 + 0.0704i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.650 - 2.42i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.62 - 0.435i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.60 - 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.01 - 3.80i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.67 + 2.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (8.71 - 2.33i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.882 + 3.29i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.762 - 0.762i)T + 41iT^{2} \)
43 \( 1 - 5.62iT - 43T^{2} \)
47 \( 1 + (-6.27 - 1.68i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.358 + 0.620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.53 + 9.44i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.943 + 0.544i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.569 - 2.12i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.04 - 6.04i)T - 71iT^{2} \)
73 \( 1 + (-1.06 - 3.98i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.96 + 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 + (-12.1 - 3.25i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.84 - 4.84i)T + 97iT^{2} \)
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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

−12.48219487933152331310713071685, −10.84812220364616324785210911030, −10.31332659731948841219340490731, −9.474675865923044816887874653477, −8.572820187217108289199772146169, −7.45349668214740084731006537991, −6.30246666845283171537347462418, −5.04649641923006313332757820957, −3.69562305422240758922884584194, −2.53534622241538702725750566302, 0.48581256505998033039703317823, 2.92444115686930856256005051381, 4.26465605136448478838013366390, 5.14395281804218229905612416640, 7.00445333596168634231222907308, 7.70727466448906024826955694734, 9.014977862640256998305043311652, 9.144276465411848070804420567439, 10.36896863503539109244005570575, 11.91267032811093990206526403172

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