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Properties

Label 2-273-21.20-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.741 - 0.671i$
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  + 2.62i·2-s + (−0.518 − 1.65i)3-s − 4.91·4-s + 2.48·5-s + (4.34 − 1.36i)6-s + (1.33 + 2.28i)7-s − 7.67i·8-s + (−2.46 + 1.71i)9-s + 6.54i·10-s + 5.20i·11-s + (2.54 + 8.12i)12-s + i·13-s + (−6.00 + 3.52i)14-s + (−1.28 − 4.11i)15-s + 10.3·16-s + 2.44·17-s + ⋯
L(s)  = 1  + 1.85i·2-s + (−0.299 − 0.954i)3-s − 2.45·4-s + 1.11·5-s + (1.77 − 0.556i)6-s + (0.506 + 0.862i)7-s − 2.71i·8-s + (−0.820 + 0.570i)9-s + 2.07i·10-s + 1.56i·11-s + (0.735 + 2.34i)12-s + 0.277i·13-s + (−1.60 + 0.941i)14-s + (−0.333 − 1.06i)15-s + 2.58·16-s + 0.592·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.741 - 0.671i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.741 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420160 + 1.08971i\)
\(L(\frac12)\) \(\approx\) \(0.420160 + 1.08971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.518 + 1.65i)T \)
7 \( 1 + (-1.33 - 2.28i)T \)
13 \( 1 - iT \)
good2 \( 1 - 2.62iT - 2T^{2} \)
5 \( 1 - 2.48T + 5T^{2} \)
11 \( 1 - 5.20iT - 11T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 - 5.95iT - 23T^{2} \)
29 \( 1 + 2.80iT - 29T^{2} \)
31 \( 1 + 6.81iT - 31T^{2} \)
37 \( 1 - 3.17T + 37T^{2} \)
41 \( 1 - 0.133T + 41T^{2} \)
43 \( 1 - 6.61T + 43T^{2} \)
47 \( 1 + 4.15T + 47T^{2} \)
53 \( 1 + 7.32iT - 53T^{2} \)
59 \( 1 + 0.723T + 59T^{2} \)
61 \( 1 + 8.23iT - 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 - 7.17iT - 71T^{2} \)
73 \( 1 + 16.4iT - 73T^{2} \)
79 \( 1 + 0.0675T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−12.70388302969083466641757387332, −11.60675511485376786874838510984, −9.778462252563263871644372764508, −9.201765438534536052530058831099, −7.968381629416921587448663445945, −7.31613084911418718762922447782, −6.26520071775545435840541535295, −5.61129842242400173383013706925, −4.75915442137652457408747751408, −1.98789975096942933341602657120, 1.04984495575005607094963841741, 2.85325759218957105125013545607, 3.86806828482618029234869290869, 5.01730162064912953599051564974, 5.97267444673705546510406292783, 8.328342613556172606308873812993, 9.128182635236845223401210386240, 10.16037951425134753624718239707, 10.56086207611025840476967247992, 11.19884942371738417113109641241

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