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Properties

Label 2-273-91.33-c1-0-4
Degree $2$
Conductor $273$
Sign $0.540 - 0.841i$
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  + (0.157 − 0.588i)2-s + i·3-s + (1.41 + 0.814i)4-s + (0.529 + 1.97i)5-s + (0.588 + 0.157i)6-s + (−2.23 + 1.40i)7-s + (1.56 − 1.56i)8-s − 9-s + 1.24·10-s + (0.0718 − 0.0718i)11-s + (−0.814 + 1.41i)12-s + (−3.09 + 1.85i)13-s + (0.476 + 1.53i)14-s + (−1.97 + 0.529i)15-s + (0.955 + 1.65i)16-s + (1.81 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.111 − 0.416i)2-s + 0.577i·3-s + (0.705 + 0.407i)4-s + (0.236 + 0.884i)5-s + (0.240 + 0.0643i)6-s + (−0.846 + 0.532i)7-s + (0.552 − 0.552i)8-s − 0.333·9-s + 0.394·10-s + (0.0216 − 0.0216i)11-s + (−0.235 + 0.407i)12-s + (−0.858 + 0.513i)13-s + (0.127 + 0.411i)14-s + (−0.510 + 0.136i)15-s + (0.238 + 0.413i)16-s + (0.440 − 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.540 - 0.841i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31325 + 0.717170i\)
\(L(\frac12)\) \(\approx\) \(1.31325 + 0.717170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.23 - 1.40i)T \)
13 \( 1 + (3.09 - 1.85i)T \)
good2 \( 1 + (-0.157 + 0.588i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.529 - 1.97i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0718 + 0.0718i)T - 11iT^{2} \)
17 \( 1 + (-1.81 + 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.02 - 1.02i)T - 19iT^{2} \)
23 \( 1 + (-4.16 + 2.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.79 + 6.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.47 - 1.73i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.94 - 0.788i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.872 - 3.25i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.21 - 4.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.97 - 0.529i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.27 + 9.14i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.16 - 0.581i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 5.95iT - 61T^{2} \)
67 \( 1 + (4.43 + 4.43i)T + 67iT^{2} \)
71 \( 1 + (0.0733 - 0.273i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.11 + 4.17i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.01 - 8.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.74 - 3.74i)T - 83iT^{2} \)
89 \( 1 + (2.10 - 7.85i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (16.5 + 4.43i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.90562801376738985978851358108, −11.18392436096943346914416527264, −10.09408188068790004491253451999, −9.662834329290720831645448424423, −8.200013436821839668924327089393, −6.88826494290116984654426669745, −6.30284380282536115183042879173, −4.67683642743129945206826204787, −3.15615498233263845602134778162, −2.55430801972867127521205386125, 1.22416955358290875311378235961, 2.92607064790278203389798407828, 4.82895219303694155046164585522, 5.83683656177834001299794659525, 6.81536329519697141992474080691, 7.60120421830077377617148335708, 8.758875821373883174891710964172, 9.942131184814228960290285440920, 10.71517445231759663723659501394, 11.98940125316577028632904582160

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