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Properties

Label 2-273-273.158-c1-0-15
Degree $2$
Conductor $273$
Sign $0.786 + 0.617i$
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  − 1.73·3-s + (−1.73 + i)4-s + (0.866 − 2.5i)7-s + 2.99·9-s + (2.99 − 1.73i)12-s + (3.46 − i)13-s + (1.99 − 3.46i)16-s + (4.83 − 4.83i)19-s + (−1.49 + 4.33i)21-s + (4.33 + 2.5i)25-s − 5.19·27-s + (1.00 + 5.19i)28-s + (−2.86 − 10.6i)31-s + (−5.19 + 2.99i)36-s + (−11.0 + 2.96i)37-s + ⋯
L(s)  = 1  − 1.00·3-s + (−0.866 + 0.5i)4-s + (0.327 − 0.944i)7-s + 0.999·9-s + (0.866 − 0.499i)12-s + (0.960 − 0.277i)13-s + (0.499 − 0.866i)16-s + (1.10 − 1.10i)19-s + (−0.327 + 0.944i)21-s + (0.866 + 0.5i)25-s − 1.00·27-s + (0.188 + 0.981i)28-s + (−0.514 − 1.92i)31-s + (−0.866 + 0.499i)36-s + (−1.81 + 0.487i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749937 - 0.259053i\)
\(L(\frac12)\) \(\approx\) \(0.749937 - 0.259053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
7 \( 1 + (-0.866 + 2.5i)T \)
13 \( 1 + (-3.46 + i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
5 \( 1 + (-4.33 - 2.5i)T^{2} \)
11 \( 1 - 11iT^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.83 + 4.83i)T - 19iT^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.86 + 10.6i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (11.0 - 2.96i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9 - 5.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 + (-0.562 + 0.562i)T - 67iT^{2} \)
71 \( 1 + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.63 + 0.705i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.06 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.42 + 16.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.64996295398045131379231857206, −11.00616327066053500289832374302, −9.999641889628206096775105757934, −9.010362177127423118063092174549, −7.77312291924538248562853475793, −6.96147991899138037417452249385, −5.54543726325998252814720280880, −4.60554796710661485156932875889, −3.57770513903349666813018849959, −0.851910396260609784028907903638, 1.39961560219702737429592142104, 3.80423190773104554637021019618, 5.18031596331125900166035076671, 5.64738131824238056118633075569, 6.83762142493620906227815171043, 8.365634259790819036733879139932, 9.161029089044769560269067254618, 10.25026962185059283096290603604, 10.98544474814420628248582250464, 12.11607845841374911950234250385

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