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Properties

Label 2-273-91.47-c1-0-16
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

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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: $\chi_{273} (229, \cdot )$
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 \( 1 + (6.89 - 6.89i)T - 97iT^{2} \)
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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

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

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