↑

Properties

Label 2-273-91.20-c1-0-13
Degree $2$
Conductor $273$
Sign $0.150 + 0.988i$
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  + (0.5 − 0.133i)2-s + (−0.866 − 0.5i)3-s + (−1.5 + 0.866i)4-s + (1.36 − 1.36i)5-s + (−0.5 − 0.133i)6-s + (0.866 − 2.5i)7-s + (−1.36 + 1.36i)8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−1 − 3.73i)11-s + 1.73·12-s + (3.23 − 1.59i)13-s + (0.0980 − 1.36i)14-s + (−1.86 + 0.499i)15-s + (1.23 − 2.13i)16-s + (−2.59 − 4.5i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.0947i)2-s + (−0.499 − 0.288i)3-s + (−0.750 + 0.433i)4-s + (0.610 − 0.610i)5-s + (−0.204 − 0.0546i)6-s + (0.327 − 0.944i)7-s + (−0.482 + 0.482i)8-s + (0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.301 − 1.12i)11-s + 0.499·12-s + (0.896 − 0.443i)13-s + (0.0262 − 0.365i)14-s + (−0.481 + 0.129i)15-s + (0.308 − 0.533i)16-s + (−0.630 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.150 + 0.988i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.150 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865324 - 0.743232i\)
\(L(\frac12)\) \(\approx\) \(0.865324 - 0.743232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
13 \( 1 + (-3.23 + 1.59i)T \)
good2 \( 1 + (-0.5 + 0.133i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.36 + 1.36i)T - 5iT^{2} \)
11 \( 1 + (1 + 3.73i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.73 + 0.732i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.36 - 3.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.26 - 3.26i)T - 31iT^{2} \)
37 \( 1 + (-2.5 - 9.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.40 - 8.96i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.53 + 2.53i)T + 47iT^{2} \)
53 \( 1 - 7.73T + 53T^{2} \)
59 \( 1 + (0.0980 - 0.366i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.59 + 3.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 - 2.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.46 + 5.46i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.29 - 8.29i)T + 73iT^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + (-6.92 + 6.92i)T - 83iT^{2} \)
89 \( 1 + (9.83 - 2.63i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.09 + 1.36i)T + (84.0 + 48.5i)T^{2} \)
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

−11.61065634180625987146011651681, −11.02345750029088713439116349901, −9.779237493133951651183452761600, −8.710119611971559169287174154162, −7.956193887299941229098777127405, −6.55913491159693394578312649433, −5.36988408368673516181525594779, −4.61200138536940701450318660164, −3.19931871238343945130541265753, −0.910902436428917534686185350642, 2.07633351458898133562482952634, 3.99595142084720817810288466031, 5.05611033501172210893795512393, 5.98405662877678639751543767251, 6.76894407315084252016827220238, 8.595412920257115749384439347850, 9.251112865754012400866823475307, 10.41150671034223228504284330400, 10.90406076188703990939500408296, 12.37416477395361882803499923851

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