↑

Properties

Label 2-273-91.31-c1-0-13
Degree $2$
Conductor $273$
Sign $0.739 - 0.672i$
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  + (2.11 + 0.566i)2-s + (0.866 + 0.5i)3-s + (2.42 + 1.39i)4-s + (0.169 − 0.631i)5-s + (1.54 + 1.54i)6-s + (−1.81 + 1.92i)7-s + (1.23 + 1.23i)8-s + (0.499 + 0.866i)9-s + (0.715 − 1.23i)10-s + (−2.46 + 0.659i)11-s + (1.39 + 2.42i)12-s + (1.72 − 3.16i)13-s + (−4.92 + 3.05i)14-s + (0.462 − 0.462i)15-s + (−0.886 − 1.53i)16-s + (2.99 − 5.18i)17-s + ⋯
L(s)  = 1  + (1.49 + 0.400i)2-s + (0.499 + 0.288i)3-s + (1.21 + 0.699i)4-s + (0.0756 − 0.282i)5-s + (0.632 + 0.632i)6-s + (−0.684 + 0.728i)7-s + (0.435 + 0.435i)8-s + (0.166 + 0.288i)9-s + (0.226 − 0.392i)10-s + (−0.741 + 0.198i)11-s + (0.403 + 0.699i)12-s + (0.478 − 0.877i)13-s + (−1.31 + 0.815i)14-s + (0.119 − 0.119i)15-s + (−0.221 − 0.383i)16-s + (0.725 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.739 - 0.672i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.739 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67786 + 1.03581i\)
\(L(\frac12)\) \(\approx\) \(2.67786 + 1.03581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 - 1.92i)T \)
13 \( 1 + (-1.72 + 3.16i)T \)
good2 \( 1 + (-2.11 - 0.566i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.169 + 0.631i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.46 - 0.659i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.99 + 5.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.456 + 1.70i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.55 - 2.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0763T + 29T^{2} \)
31 \( 1 + (7.61 - 2.04i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.86 - 10.6i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.05 - 1.05i)T + 41iT^{2} \)
43 \( 1 - 0.901iT - 43T^{2} \)
47 \( 1 + (-3.26 - 0.875i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.696 + 1.20i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 + 5.09i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.64 + 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.940 - 3.51i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.48 + 4.48i)T - 71iT^{2} \)
73 \( 1 + (-4.00 - 14.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.60 - 9.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.20 + 6.20i)T + 83iT^{2} \)
89 \( 1 + (-8.92 - 2.39i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.89 + 6.89i)T + 97iT^{2} \)
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

−12.45234206944165009582533319669, −11.43630311452482716287493265431, −10.05529321256204403944141616273, −9.177501308992358887654206463313, −7.940122631171156963848408396570, −6.84675242105559195778198245063, −5.54849075159934244722378222506, −5.06661199001711636259607916869, −3.52137113112731731481839629997, −2.75252179035784332919589542291, 2.10483914426775793746181541376, 3.47197158106133007520661534174, 4.10931642920899424562981923299, 5.70225807168978520241669312187, 6.51549755789687756524402799049, 7.65936061896552539212994389696, 8.929146495498457151359280137103, 10.33736469656054199167241581443, 10.90907599398529369038898760064, 12.27931056248929240536810796892

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