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Properties

Label 2-273-91.51-c1-0-12
Degree $2$
Conductor $273$
Sign $-0.172 + 0.985i$
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  + (−0.161 − 0.0930i)2-s + (−0.5 − 0.866i)3-s + (−0.982 − 1.70i)4-s + (2.28 + 1.31i)5-s + 0.186i·6-s + (−0.161 − 2.64i)7-s + 0.738i·8-s + (−0.499 + 0.866i)9-s + (−0.245 − 0.425i)10-s + (1.56 − 0.904i)11-s + (−0.982 + 1.70i)12-s + (−2.45 − 2.63i)13-s + (−0.219 + 0.440i)14-s − 2.63i·15-s + (−1.89 + 3.28i)16-s + (−3.13 − 5.42i)17-s + ⋯
L(s)  = 1  + (−0.113 − 0.0658i)2-s + (−0.288 − 0.499i)3-s + (−0.491 − 0.851i)4-s + (1.02 + 0.590i)5-s + 0.0759i·6-s + (−0.0609 − 0.998i)7-s + 0.260i·8-s + (−0.166 + 0.288i)9-s + (−0.0776 − 0.134i)10-s + (0.472 − 0.272i)11-s + (−0.283 + 0.491i)12-s + (−0.681 − 0.731i)13-s + (−0.0587 + 0.117i)14-s − 0.681i·15-s + (−0.474 + 0.821i)16-s + (−0.760 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.172 + 0.985i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.172 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681953 - 0.811699i\)
\(L(\frac12)\) \(\approx\) \(0.681953 - 0.811699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.161 + 2.64i)T \)
13 \( 1 + (2.45 + 2.63i)T \)
good2 \( 1 + (0.161 + 0.0930i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.28 - 1.31i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.56 + 0.904i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.13 + 5.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.44 - 1.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.456T + 29T^{2} \)
31 \( 1 + (2.76 - 1.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.00 - 1.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.18iT - 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + (-10.3 - 5.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.23 - 3.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.83 + 5.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.59 - 7.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.40 + 0.810i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.85iT - 71T^{2} \)
73 \( 1 + (12.3 - 7.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.86 + 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + (-0.0791 - 0.0457i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.62iT - 97T^{2} \)
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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

−11.38460776189137719221013243338, −10.57169895805471864620752231060, −9.899819341862937117192457199539, −9.043145084260193980209445405143, −7.50187087631272249384136793177, −6.59542295737372113819742816459, −5.69139004828994887837586749635, −4.58061817983862290583126189253, −2.61622183842162789022455019420, −0.930193756395866512996321885256, 2.19293243324152481574953170648, 3.91699476580815451751786696175, 5.03684403199948280665493065227, 5.97620188810023380151006456350, 7.28583416078969232689177064545, 8.892725533915744038011774302390, 9.053677027621169649692859134567, 9.956178890692108514695994018885, 11.38568699441813832964378425071, 12.25808466975861862139132130374

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