↑

Properties

Label 2-273-91.74-c1-0-5
Degree $2$
Conductor $273$
Sign $0.923 + 0.383i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.65·2-s + (−0.5 − 0.866i)3-s + 0.744·4-s + (1.05 + 1.81i)5-s + (0.828 + 1.43i)6-s + (−1.49 − 2.18i)7-s + 2.07·8-s + (−0.499 + 0.866i)9-s + (−1.73 − 3.01i)10-s + (0.152 + 0.263i)11-s + (−0.372 − 0.644i)12-s + (−0.494 + 3.57i)13-s + (2.47 + 3.61i)14-s + (1.05 − 1.81i)15-s − 4.93·16-s + 5.81·17-s + ⋯
L(s)  = 1  − 1.17·2-s + (−0.288 − 0.499i)3-s + 0.372·4-s + (0.469 + 0.813i)5-s + (0.338 + 0.585i)6-s + (−0.564 − 0.825i)7-s + 0.735·8-s + (−0.166 + 0.288i)9-s + (−0.550 − 0.952i)10-s + (0.0458 + 0.0794i)11-s + (−0.107 − 0.186i)12-s + (−0.137 + 0.990i)13-s + (0.660 + 0.967i)14-s + (0.271 − 0.469i)15-s − 1.23·16-s + 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633359 - 0.126253i\)
\(L(\frac12)\) \(\approx\) \(0.633359 - 0.126253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 + 2.18i)T \)
13 \( 1 + (0.494 - 3.57i)T \)
good2 \( 1 + 1.65T + 2T^{2} \)
5 \( 1 + (-1.05 - 1.81i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.152 - 0.263i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.81T + 17T^{2} \)
19 \( 1 + (-3.74 + 6.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.01T + 23T^{2} \)
29 \( 1 + (-1.09 + 1.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.51 + 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.104T + 37T^{2} \)
41 \( 1 + (1.00 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.03 - 6.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.30 - 7.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.10 - 1.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + (-5.51 + 9.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.04 + 8.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.149 + 0.259i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.96 - 5.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.54 - 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.33T + 83T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 + (3.19 + 5.53i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.43634361563712440845252581147, −10.77293921680829533177992636114, −9.774410341129119986245349223912, −9.292510588592844240269490910626, −7.77828867980455309015394325510, −7.12443335568443257507370301146, −6.32672630236845410981386886977, −4.64888600547194367302684702461, −2.83040184012263652371514879447, −1.02383815149656513418117751200, 1.16354847254692681478832615528, 3.26880933353759496218505764252, 5.11900518054846839754780609374, 5.72584958106923039095444718080, 7.36987821096012695644132628621, 8.498458921210606170143005154000, 9.122832944712634465673534086530, 10.00079161027396853653410699099, 10.47440520730955204757630446569, 11.98388965092184577763340223759

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