Properties

Label 2-273-91.31-c1-0-12
Degree $2$
Conductor $273$
Sign $0.183 + 0.983i$
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  + (−1.28 − 0.345i)2-s + (0.866 + 0.5i)3-s + (−0.191 − 0.110i)4-s + (0.306 − 1.14i)5-s + (−0.943 − 0.943i)6-s + (1.14 − 2.38i)7-s + (2.09 + 2.09i)8-s + (0.499 + 0.866i)9-s + (−0.791 + 1.37i)10-s + (−2.90 + 0.778i)11-s + (−0.110 − 0.191i)12-s + (−0.759 − 3.52i)13-s + (−2.30 + 2.67i)14-s + (0.838 − 0.838i)15-s + (−1.75 − 3.03i)16-s + (0.562 − 0.974i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.244i)2-s + (0.499 + 0.288i)3-s + (−0.0955 − 0.0551i)4-s + (0.137 − 0.512i)5-s + (−0.385 − 0.385i)6-s + (0.433 − 0.901i)7-s + (0.740 + 0.740i)8-s + (0.166 + 0.288i)9-s + (−0.250 + 0.433i)10-s + (−0.876 + 0.234i)11-s + (−0.0318 − 0.0551i)12-s + (−0.210 − 0.977i)13-s + (−0.615 + 0.715i)14-s + (0.216 − 0.216i)15-s + (−0.438 − 0.759i)16-s + (0.136 − 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.183 + 0.983i$
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: \((2,\ 273,\ (\ :1/2),\ 0.183 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652046 - 0.541627i\)
\(L(\frac12)\) \(\approx\) \(0.652046 - 0.541627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 + 2.38i)T \)
13 \( 1 + (0.759 + 3.52i)T \)
good2 \( 1 + (1.28 + 0.345i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.306 + 1.14i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.90 - 0.778i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.562 + 0.974i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.70 + 6.38i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.76 + 2.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.66T + 29T^{2} \)
31 \( 1 + (-2.53 + 0.678i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.56 + 9.55i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.47 - 3.47i)T + 41iT^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (4.00 + 1.07i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.28 - 7.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.43 - 9.06i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (8.03 - 4.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.888 - 3.31i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.61 + 7.61i)T - 71iT^{2} \)
73 \( 1 + (-2.84 - 10.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0499 - 0.0499i)T + 83iT^{2} \)
89 \( 1 + (-13.6 - 3.65i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.73 + 2.73i)T + 97iT^{2} \)
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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

−11.10985584626720381790609006532, −10.69011555606826463648216868558, −9.650533836986243191837682868949, −9.009712914297705683174648332638, −7.928995469751124636655594594536, −7.35626710884783612031503497265, −5.24324360054843077028805494952, −4.56677975300339648637275305657, −2.71819928711913916444801232519, −0.896975502107620248445107183374, 1.86272218198272257091565986071, 3.40468660550616814412056841882, 5.04422578668322927705413852960, 6.45464129341525804301880670552, 7.58304689639274523624638541260, 8.250515607343314205883486922319, 9.137933476261036953982769171286, 9.927530406485593594335298179750, 10.97524689716035271433522138977, 12.11148016516484625159070049450

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