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Properties

Label 2-273-13.3-c1-0-10
Degree $2$
Conductor $273$
Sign $-0.982 - 0.186i$
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  + (−0.766 − 1.32i)2-s + (−0.5 − 0.866i)3-s + (−0.173 + 0.300i)4-s + 0.120·5-s + (−0.766 + 1.32i)6-s + (0.5 − 0.866i)7-s − 2.53·8-s + (−0.499 + 0.866i)9-s + (−0.0923 − 0.160i)10-s + (−2.28 − 3.96i)11-s + 0.347·12-s + (−0.245 − 3.59i)13-s − 1.53·14-s + (−0.0603 − 0.104i)15-s + (2.28 + 3.96i)16-s + (−2.46 + 4.26i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.938i)2-s + (−0.288 − 0.499i)3-s + (−0.0868 + 0.150i)4-s + 0.0539·5-s + (−0.312 + 0.541i)6-s + (0.188 − 0.327i)7-s − 0.895·8-s + (−0.166 + 0.288i)9-s + (−0.0292 − 0.0506i)10-s + (−0.689 − 1.19i)11-s + 0.100·12-s + (−0.0679 − 0.997i)13-s − 0.409·14-s + (−0.0155 − 0.0269i)15-s + (0.571 + 0.990i)16-s + (−0.596 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.982 - 0.186i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.982 - 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0626691 + 0.667024i\)
\(L(\frac12)\) \(\approx\) \(0.0626691 + 0.667024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.245 + 3.59i)T \)
good2 \( 1 + (0.766 + 1.32i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.120T + 5T^{2} \)
11 \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.46 - 4.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.379 + 0.657i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.73 - 4.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.33 + 9.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.63T + 31T^{2} \)
37 \( 1 + (-2.35 - 4.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.45 + 9.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.631 - 1.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.94T + 47T^{2} \)
53 \( 1 - 9.49T + 53T^{2} \)
59 \( 1 + (-4.49 + 7.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.72 + 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.51 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.23 - 2.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.89T + 73T^{2} \)
79 \( 1 + 8.18T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + (-3.37 - 5.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.03 + 8.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.27143045029076838589433520546, −10.61678373188985936936267525034, −9.806633553299340532621153820272, −8.512301138808216115203163499187, −7.79353627891015313505478129834, −6.25542739924107470334920201438, −5.47706293428045336316904736056, −3.55022963627501722880138021158, −2.21902404294144929230327331910, −0.61015986673635251661838547009, 2.57386335007314017258916863618, 4.43076219582706695934767763619, 5.45010986472568899679325687920, 6.72426720452644624850013339917, 7.38440704250593205996289369893, 8.627120743398504793872872552832, 9.348602860207039164061790829424, 10.25575015738963350761554604071, 11.54460293483440409317698619804, 12.16874894512683366762274451025

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