Properties

Label 2-273-273.116-c2-0-48
Degree $2$
Conductor $273$
Sign $0.949 + 0.313i$
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  + (1.70 + 2.96i)2-s + (−2.79 + 1.09i)3-s + (−3.84 + 6.66i)4-s + (−3.81 − 6.60i)5-s + (−8.01 − 6.39i)6-s + (4.11 − 5.66i)7-s − 12.6·8-s + (6.59 − 6.11i)9-s + (13.0 − 22.5i)10-s + (1.31 − 2.27i)11-s + (3.44 − 22.8i)12-s − 13·13-s + (23.8 + 2.51i)14-s + (17.8 + 14.2i)15-s + (−6.19 − 10.7i)16-s + (−23.8 − 13.7i)17-s + ⋯
L(s)  = 1  + (0.854 + 1.48i)2-s + (−0.930 + 0.365i)3-s + (−0.961 + 1.66i)4-s + (−0.762 − 1.32i)5-s + (−1.33 − 1.06i)6-s + (0.588 − 0.808i)7-s − 1.57·8-s + (0.733 − 0.679i)9-s + (1.30 − 2.25i)10-s + (0.119 − 0.207i)11-s + (0.286 − 1.90i)12-s − 13-s + (1.70 + 0.179i)14-s + (1.19 + 0.951i)15-s + (−0.386 − 0.670i)16-s + (−1.40 − 0.808i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ 0.949 + 0.313i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.967743 - 0.155415i\)
\(L(\frac12)\) \(\approx\) \(0.967743 - 0.155415i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.79 - 1.09i)T \)
7 \( 1 + (-4.11 + 5.66i)T \)
13 \( 1 + 13T \)
good2 \( 1 + (-1.70 - 2.96i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (3.81 + 6.60i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.31 + 2.27i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + (23.8 + 13.7i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.27 - 0.736i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-28.1 + 16.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 38.7iT - 841T^{2} \)
31 \( 1 + (13.6 + 7.86i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5.09 - 2.94i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 47.3T + 1.68e3T^{2} \)
43 \( 1 + 12.6T + 1.84e3T^{2} \)
47 \( 1 + (11.4 + 19.7i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (78.1 + 45.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-18.1 + 31.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-32.6 - 56.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.13 - 4.69i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 94.6T + 5.04e3T^{2} \)
73 \( 1 + (-92.0 - 53.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (18.4 + 31.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 11.2T + 6.88e3T^{2} \)
89 \( 1 + (-77.0 - 133. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 22.6iT - 9.40e3T^{2} \)
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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

−11.84067254230080962466466364897, −11.01492715587685133642387832025, −9.442770113016010785690539807015, −8.392739450111161653389558331757, −7.42502868809712515644125229787, −6.64054022881733480953926854283, −5.24101985093474387843751979816, −4.65491733028823643406926103265, −4.11113267120683561603799271543, −0.42205917276074737879089976426, 1.84004976033896518960034342342, 2.96846380773337638461589843288, 4.37257366889860603305495645210, 5.26737974221269035258213771823, 6.58142544727591957876076014194, 7.57814881969074141395177360819, 9.277876425487439927783307746387, 10.67925597453669682887872503875, 10.95900412590448677813704024555, 11.64599728223421064289441591711

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