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Properties

Label 2-273-91.81-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.866 + 0.498i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 − 2.20i)2-s + 3-s + (−2.25 − 3.90i)4-s + (−1.40 − 2.44i)5-s + (1.27 − 2.20i)6-s + (−0.0337 + 2.64i)7-s − 6.39·8-s + 9-s − 7.18·10-s + 0.531·11-s + (−2.25 − 3.90i)12-s + (2.71 + 2.36i)13-s + (5.80 + 3.44i)14-s + (−1.40 − 2.44i)15-s + (−3.64 + 6.31i)16-s + (−0.202 − 0.351i)17-s + ⋯
L(s)  = 1  + (0.901 − 1.56i)2-s + 0.577·3-s + (−1.12 − 1.95i)4-s + (−0.630 − 1.09i)5-s + (0.520 − 0.901i)6-s + (−0.0127 + 0.999i)7-s − 2.26·8-s + 0.333·9-s − 2.27·10-s + 0.160·11-s + (−0.650 − 1.12i)12-s + (0.753 + 0.657i)13-s + (1.55 + 0.921i)14-s + (−0.363 − 0.630i)15-s + (−0.911 + 1.57i)16-s + (−0.0492 − 0.0852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.866 + 0.498i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.866 + 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525310 - 1.96602i\)
\(L(\frac12)\) \(\approx\) \(0.525310 - 1.96602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (0.0337 - 2.64i)T \)
13 \( 1 + (-2.71 - 2.36i)T \)
good2 \( 1 + (-1.27 + 2.20i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.40 + 2.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.531T + 11T^{2} \)
17 \( 1 + (0.202 + 0.351i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.67T + 19T^{2} \)
23 \( 1 + (-1.27 + 2.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.44 + 7.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.44 - 7.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.47 + 7.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.44 - 9.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.52 - 7.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.45 - 7.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.51 - 6.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0364 + 0.0631i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.74T + 61T^{2} \)
67 \( 1 - 7.96T + 67T^{2} \)
71 \( 1 + (0.535 - 0.928i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.729 + 1.26i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.53 - 6.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.449T + 83T^{2} \)
89 \( 1 + (-0.461 + 0.799i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0841 - 0.145i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.65648825084692444029518742447, −11.02292837785769936934862189793, −9.440835174295761408928090757368, −9.142958804149632001933685549548, −7.980210661136511261216061963307, −5.98609284880407188535587846183, −4.82822103773292254524049816349, −4.00297806871824021210119016665, −2.79872242060875363129434845830, −1.38996193173241685078806458794, 3.41290111510347647346235607714, 3.81858685557187368807502000890, 5.32290298845259193447819427635, 6.58369901322561590776435770011, 7.38344398122879257232769469533, 7.80359000314626740180174058011, 9.026080509650121521005134345720, 10.45821047592643814525182493093, 11.44345616775643393257108567426, 12.83489891701929856556510488929

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