Properties

Label 2-273-91.31-c1-0-10
Degree $2$
Conductor $273$
Sign $0.727 - 0.686i$
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  + (2.20 + 0.590i)2-s + (−0.866 − 0.5i)3-s + (2.77 + 1.60i)4-s + (−0.460 + 1.71i)5-s + (−1.61 − 1.61i)6-s + (2.30 + 1.30i)7-s + (1.93 + 1.93i)8-s + (0.499 + 0.866i)9-s + (−2.02 + 3.51i)10-s + (1.75 − 0.470i)11-s + (−1.60 − 2.77i)12-s + (−2.71 − 2.37i)13-s + (4.30 + 4.23i)14-s + (1.25 − 1.25i)15-s + (−0.0761 − 0.131i)16-s + (−0.854 + 1.47i)17-s + ⋯
L(s)  = 1  + (1.55 + 0.417i)2-s + (−0.499 − 0.288i)3-s + (1.38 + 0.800i)4-s + (−0.206 + 0.768i)5-s + (−0.658 − 0.658i)6-s + (0.870 + 0.492i)7-s + (0.685 + 0.685i)8-s + (0.166 + 0.288i)9-s + (−0.641 + 1.11i)10-s + (0.529 − 0.141i)11-s + (−0.462 − 0.800i)12-s + (−0.751 − 0.659i)13-s + (1.14 + 1.13i)14-s + (0.324 − 0.324i)15-s + (−0.0190 − 0.0329i)16-s + (−0.207 + 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.727 - 0.686i$
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: \((2,\ 273,\ (\ :1/2),\ 0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31529 + 0.919780i\)
\(L(\frac12)\) \(\approx\) \(2.31529 + 0.919780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 - 1.30i)T \)
13 \( 1 + (2.71 + 2.37i)T \)
good2 \( 1 + (-2.20 - 0.590i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.460 - 1.71i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.75 + 0.470i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.854 - 1.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.28 + 4.80i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.835 - 0.482i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.21T + 29T^{2} \)
31 \( 1 + (7.81 - 2.09i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.40 + 5.25i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.11 + 4.11i)T + 41iT^{2} \)
43 \( 1 + 5.65iT - 43T^{2} \)
47 \( 1 + (-2.68 - 0.718i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.808 - 1.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.20 - 11.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.12 + 3.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.142 - 0.531i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.5 - 11.5i)T - 71iT^{2} \)
73 \( 1 + (1.96 + 7.31i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.06 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.42 - 9.42i)T + 83iT^{2} \)
89 \( 1 + (-12.4 - 3.32i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.94 - 1.94i)T + 97iT^{2} \)
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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

−12.14719235321487218490365532818, −11.39579798235102588386653125842, −10.69866209838541715869827823687, −9.031708526819165751110207994994, −7.49437611796630189120881052526, −6.94540544641941812719892716375, −5.72109206357129667226943118926, −5.06807534380866650097538345267, −3.78635078127597525705200672105, −2.42600556292429085238413347487, 1.75435651860778826346230254117, 3.71908728546971422739489923912, 4.60075656087540383557617879442, 5.18956129413616633696723538912, 6.41562662011677552407372055000, 7.65931318580823260544838541080, 9.062300947989921858701852038544, 10.26569313125569746000793716855, 11.41190195205410674498061582855, 11.79964631709313707232140952124

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