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Properties

Label 2-273-91.47-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.402 - 0.915i$
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  + (−2.05 + 0.549i)2-s + (−0.866 + 0.5i)3-s + (2.17 − 1.25i)4-s + (1.02 + 3.81i)5-s + (1.50 − 1.50i)6-s + (2.63 + 0.231i)7-s + (−0.766 + 0.766i)8-s + (0.499 − 0.866i)9-s + (−4.19 − 7.25i)10-s + (1.50 + 0.403i)11-s + (−1.25 + 2.17i)12-s + (2.17 − 2.87i)13-s + (−5.53 + 0.973i)14-s + (−2.79 − 2.79i)15-s + (−1.35 + 2.35i)16-s + (1.00 + 1.73i)17-s + ⋯
L(s)  = 1  + (−1.45 + 0.388i)2-s + (−0.499 + 0.288i)3-s + (1.08 − 0.627i)4-s + (0.456 + 1.70i)5-s + (0.613 − 0.613i)6-s + (0.996 + 0.0875i)7-s + (−0.270 + 0.270i)8-s + (0.166 − 0.288i)9-s + (−1.32 − 2.29i)10-s + (0.454 + 0.121i)11-s + (−0.362 + 0.627i)12-s + (0.603 − 0.797i)13-s + (−1.47 + 0.260i)14-s + (−0.720 − 0.720i)15-s + (−0.339 + 0.588i)16-s + (0.243 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354102 + 0.542518i\)
\(L(\frac12)\) \(\approx\) \(0.354102 + 0.542518i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.231i)T \)
13 \( 1 + (-2.17 + 2.87i)T \)
good2 \( 1 + (2.05 - 0.549i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.02 - 3.81i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.50 - 0.403i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.00 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.806 - 3.00i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.40 - 1.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.87T + 29T^{2} \)
31 \( 1 + (-1.40 - 0.375i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.08 + 7.76i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.75 - 6.75i)T - 41iT^{2} \)
43 \( 1 - 3.27iT - 43T^{2} \)
47 \( 1 + (0.00770 - 0.00206i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.83 + 11.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.49 + 9.32i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.86 - 4.54i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.15 - 4.31i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.217 + 0.217i)T + 71iT^{2} \)
73 \( 1 + (-1.31 + 4.92i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.44 - 9.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \)
89 \( 1 + (10.2 - 2.74i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.68 + 5.68i)T - 97iT^{2} \)
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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

−11.47428648592537261240522705439, −10.98341909161729626204133849783, −10.27235020001115687800422328935, −9.536766396847114800788568669526, −8.274132536106629754130802201767, −7.41764274903830139609101992725, −6.52667134613945453981144127057, −5.56665020838691208999568090814, −3.58961732058557003123697739374, −1.71432398915802571165468197584, 0.965977747995761669002307255676, 1.81349201436360084625625472046, 4.54563392249247279200851477628, 5.42390152928851194797430328209, 6.98383985843573338256720835565, 8.140263282184346816013033298845, 8.874438076414627266304755067774, 9.402262921141737531907750880136, 10.63752791263460046025776101964, 11.55670888015291792550921044713

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