↑

Properties

Label 2-273-273.269-c1-0-16
Degree $2$
Conductor $273$
Sign $0.654 - 0.756i$
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  − 0.0504i·2-s + (0.667 + 1.59i)3-s + 1.99·4-s + (1.19 + 2.07i)5-s + (0.0806 − 0.0336i)6-s + (0.00656 − 2.64i)7-s − 0.201i·8-s + (−2.11 + 2.13i)9-s + (0.104 − 0.0603i)10-s + (2.85 − 1.64i)11-s + (1.33 + 3.19i)12-s + (−3.26 − 1.53i)13-s + (−0.133 − 0.000331i)14-s + (−2.51 + 3.29i)15-s + 3.98·16-s − 6.34·17-s + ⋯
L(s)  = 1  − 0.0356i·2-s + (0.385 + 0.922i)3-s + 0.998·4-s + (0.534 + 0.926i)5-s + (0.0329 − 0.0137i)6-s + (0.00248 − 0.999i)7-s − 0.0712i·8-s + (−0.703 + 0.710i)9-s + (0.0330 − 0.0190i)10-s + (0.861 − 0.497i)11-s + (0.384 + 0.921i)12-s + (−0.904 − 0.427i)13-s + (−0.0356 − 8.84e−5i)14-s + (−0.648 + 0.850i)15-s + 0.996·16-s − 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.654 - 0.756i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.654 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63414 + 0.746804i\)
\(L(\frac12)\) \(\approx\) \(1.63414 + 0.746804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.667 - 1.59i)T \)
7 \( 1 + (-0.00656 + 2.64i)T \)
13 \( 1 + (3.26 + 1.53i)T \)
good2 \( 1 + 0.0504iT - 2T^{2} \)
5 \( 1 + (-1.19 - 2.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.85 + 1.64i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.34T + 17T^{2} \)
19 \( 1 + (4.30 + 2.48i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.78iT - 23T^{2} \)
29 \( 1 + (-2.15 - 1.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.40 - 2.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 + (0.726 - 1.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.16 + 3.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.7 + 6.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.83T + 59T^{2} \)
61 \( 1 + (-2.46 - 1.42i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.719 + 1.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.14 - 2.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.79 - 2.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.32 - 4.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 5.09T + 89T^{2} \)
97 \( 1 + (-11.0 + 6.38i)T + (48.5 - 84.0i)T^{2} \)
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

−11.57770942369489289210492942241, −10.89101259780616561249711044036, −10.34980032859512749998061319663, −9.449180677912405721313339989666, −8.149534231596786867874648479278, −6.91597132213815368685127856160, −6.31154267555430133739137383586, −4.66320215239932239897328392659, −3.36855516231362339784409453852, −2.31691861720708044076285189334, 1.77357394147677470240486064079, 2.51245815730282425912008762456, 4.62568688928893891181901664432, 6.17415297781559636867724833181, 6.59134006857535267865382187107, 7.924148990123702974624235970428, 8.866150387160431680976970399429, 9.567016399456887668823686360588, 11.11483208675002249871132347409, 12.18642107856765530578453181920

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