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Properties

Label 2-273-91.81-c1-0-8
Degree $2$
Conductor $273$
Sign $0.851 + 0.523i$
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  + (0.857 − 1.48i)2-s − 3-s + (−0.470 − 0.815i)4-s + (1.22 + 2.12i)5-s + (−0.857 + 1.48i)6-s + (2.18 + 1.49i)7-s + 1.81·8-s + 9-s + 4.21·10-s − 1.03·11-s + (0.470 + 0.815i)12-s + (−3.36 + 1.29i)13-s + (4.09 − 1.95i)14-s + (−1.22 − 2.12i)15-s + (2.49 − 4.32i)16-s + (−1.50 − 2.61i)17-s + ⋯
L(s)  = 1  + (0.606 − 1.05i)2-s − 0.577·3-s + (−0.235 − 0.407i)4-s + (0.549 + 0.951i)5-s + (−0.350 + 0.606i)6-s + (0.824 + 0.565i)7-s + 0.641·8-s + 0.333·9-s + 1.33·10-s − 0.313·11-s + (0.135 + 0.235i)12-s + (−0.932 + 0.360i)13-s + (1.09 − 0.523i)14-s + (−0.317 − 0.549i)15-s + (0.624 − 1.08i)16-s + (−0.366 − 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.851 + 0.523i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.851 + 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66642 - 0.471233i\)
\(L(\frac12)\) \(\approx\) \(1.66642 - 0.471233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (-2.18 - 1.49i)T \)
13 \( 1 + (3.36 - 1.29i)T \)
good2 \( 1 + (-0.857 + 1.48i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
17 \( 1 + (1.50 + 2.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + (-1.73 + 3.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.01 + 6.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 - 2.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.54 + 4.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.21 - 5.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.88 - 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.90 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.47 + 7.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.60T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + (1.63 - 2.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.50 + 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.211 + 0.366i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.34T + 83T^{2} \)
89 \( 1 + (2.65 - 4.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.92 + 5.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.75727791811565144327637818199, −11.06988109641720869103135329893, −10.34983303482888988767240937322, −9.390273648456909628436898106157, −7.75031506629496176858909592025, −6.78991853350869479604035013848, −5.40180406921861060258440844757, −4.58329449987000349139808927121, −2.93588623919755449457382933908, −1.98642670966409309136210609331, 1.53134382426272591091615554351, 4.19016429344109739918876632175, 5.29000492532547540328281400393, 5.50923821674895905144757338078, 7.06499350790696868290292066201, 7.69771458816490599056663491054, 8.958438258343625354837460042300, 10.19615906585404819626463282165, 11.01649916814851934144161700195, 12.22758448403397567826123932963

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