↑

Properties

Label 2-273-273.44-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.999 + 0.0109i$
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  + (0.965 − 0.258i)2-s + (−1.33 + 1.10i)3-s + (−0.866 + 0.5i)4-s + (−0.565 + 0.151i)5-s + (−1 + 1.41i)6-s + (−2.38 − 1.15i)7-s + (−2.12 + 2.12i)8-s + (0.548 − 2.94i)9-s + (−0.507 + 0.292i)10-s + (−5.62 − 1.50i)11-s + (0.599 − 1.62i)12-s + (2 − 3i)13-s + (−2.59 − 0.500i)14-s + (0.585 − 0.828i)15-s + (−0.500 + 0.866i)16-s + (0.914 + 1.58i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.769 + 0.639i)3-s + (−0.433 + 0.250i)4-s + (−0.253 + 0.0678i)5-s + (−0.408 + 0.577i)6-s + (−0.899 − 0.436i)7-s + (−0.749 + 0.749i)8-s + (0.182 − 0.983i)9-s + (−0.160 + 0.0926i)10-s + (−1.69 − 0.454i)11-s + (0.173 − 0.469i)12-s + (0.554 − 0.832i)13-s + (−0.694 − 0.133i)14-s + (0.151 − 0.213i)15-s + (−0.125 + 0.216i)16-s + (0.221 + 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.0109i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.999 + 0.0109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00104991 - 0.191480i\)
\(L(\frac12)\) \(\approx\) \(0.00104991 - 0.191480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.33 - 1.10i)T \)
7 \( 1 + (2.38 + 1.15i)T \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (-0.965 + 0.258i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.565 - 0.151i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.62 + 1.50i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.914 - 1.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.72 - 6.43i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.29 - 5.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - iT - 29T^{2} \)
31 \( 1 + (0.800 + 0.214i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.86 - 1.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4 + 4i)T + 41iT^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + (0.170 + 0.634i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-7.49 + 4.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.16 - 2.45i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.0 + 3.49i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.464 + 0.464i)T + 71iT^{2} \)
73 \( 1 + (3.32 - 12.3i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.707 - 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.07 + 3.07i)T + 83iT^{2} \)
89 \( 1 + (3.23 + 12.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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

−12.43320540562894458638171814628, −11.57119383493833472962820851856, −10.40762886377249407611056259630, −9.965160129257815178410135390038, −8.529392188333631294197805155892, −7.52025912310885645714557985231, −5.71244423803793057197612389813, −5.51359648396487181235693585158, −3.83215865540780670844900373813, −3.33929586598548899917009468591, 0.12393697984510844068799312871, 2.65321542423621068791012144469, 4.42202853686596351795430044587, 5.33683394231939051263006930221, 6.25382407325303925775636344855, 7.14884918315768160197965579540, 8.445804805448458410010547869229, 9.659338504810700681751786976632, 10.54375494427417887096915524687, 11.78199185006664593960586419789

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