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Properties

Label 2-273-91.69-c2-0-24
Degree $2$
Conductor $273$
Sign $0.999 - 0.0393i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.63 + 0.944i)2-s + (−1.5 − 0.866i)3-s + (−0.214 − 0.372i)4-s + 8.59·5-s + (−1.63 − 2.83i)6-s + (0.630 + 6.97i)7-s − 8.36i·8-s + (1.5 + 2.59i)9-s + (14.0 + 8.12i)10-s + (−3.14 − 1.81i)11-s + 0.744i·12-s + (0.827 − 12.9i)13-s + (−5.55 + 12.0i)14-s + (−12.8 − 7.44i)15-s + (7.04 − 12.2i)16-s + (15.2 − 8.81i)17-s + ⋯
L(s)  = 1  + (0.818 + 0.472i)2-s + (−0.5 − 0.288i)3-s + (−0.0536 − 0.0930i)4-s + 1.71·5-s + (−0.272 − 0.472i)6-s + (0.0900 + 0.995i)7-s − 1.04i·8-s + (0.166 + 0.288i)9-s + (1.40 + 0.812i)10-s + (−0.285 − 0.164i)11-s + 0.0620i·12-s + (0.0636 − 0.997i)13-s + (−0.396 + 0.857i)14-s + (−0.859 − 0.496i)15-s + (0.440 − 0.763i)16-s + (0.898 − 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0393i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ 0.999 - 0.0393i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.63571 + 0.0518150i\)
\(L(\frac12)\) \(\approx\) \(2.63571 + 0.0518150i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-0.630 - 6.97i)T \)
13 \( 1 + (-0.827 + 12.9i)T \)
good2 \( 1 + (-1.63 - 0.944i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 8.59T + 25T^{2} \)
11 \( 1 + (3.14 + 1.81i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + (-15.2 + 8.81i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-18.2 - 31.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.5 + 18.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (16.1 - 28.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 21.9T + 961T^{2} \)
37 \( 1 + (-16.5 - 9.53i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-22.9 + 39.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (18.0 + 31.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 48.6T + 2.20e3T^{2} \)
53 \( 1 + 30.8T + 2.80e3T^{2} \)
59 \( 1 + (-50.8 - 88.0i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (86.4 - 49.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (61.6 + 35.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (68.9 - 39.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + 15.3T + 5.32e3T^{2} \)
79 \( 1 + 60.8T + 6.24e3T^{2} \)
83 \( 1 - 47.1T + 6.88e3T^{2} \)
89 \( 1 + (-30.6 + 53.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-49.2 - 85.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.12472784056665660202459805264, −10.52671349568154538819109096052, −9.927064936817342571558788195488, −8.960911759105189527299893842355, −7.45145511365788490604210176910, −6.11704670705776898447533360691, −5.60055980907378165971285377630, −5.17479861618756997668534424686, −3.05733290742422227301005163309, −1.41953730067963520191097542788, 1.62592032765877364901830949583, 3.13959878669093173386661464527, 4.54442335835106231269647113583, 5.31779901918919078165718276389, 6.36516580948073009194092941965, 7.59505984441241410115352480548, 9.265093521938389240458555593843, 9.816629869939005361385481557419, 10.97039517624823449539066060232, 11.55224589805800158106674936426

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