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Properties

Label 2-273-91.45-c1-0-8
Degree $2$
Conductor $273$
Sign $0.808 - 0.588i$
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  + (−1.48 + 1.48i)2-s + (0.866 − 0.5i)3-s − 2.39i·4-s + (0.507 − 1.89i)5-s + (−0.542 + 2.02i)6-s + (−0.313 + 2.62i)7-s + (0.588 + 0.588i)8-s + (0.499 − 0.866i)9-s + (2.05 + 3.55i)10-s + (0.648 − 2.42i)11-s + (−1.19 − 2.07i)12-s + (3.33 + 1.36i)13-s + (−3.43 − 4.35i)14-s + (−0.507 − 1.89i)15-s + 3.04·16-s + 7.32·17-s + ⋯
L(s)  = 1  + (−1.04 + 1.04i)2-s + (0.499 − 0.288i)3-s − 1.19i·4-s + (0.226 − 0.846i)5-s + (−0.221 + 0.826i)6-s + (−0.118 + 0.992i)7-s + (0.207 + 0.207i)8-s + (0.166 − 0.288i)9-s + (0.649 + 1.12i)10-s + (0.195 − 0.730i)11-s + (−0.345 − 0.599i)12-s + (0.925 + 0.378i)13-s + (−0.916 − 1.16i)14-s + (−0.130 − 0.488i)15-s + 0.762·16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901497 + 0.293116i\)
\(L(\frac12)\) \(\approx\) \(0.901497 + 0.293116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.313 - 2.62i)T \)
13 \( 1 + (-3.33 - 1.36i)T \)
good2 \( 1 + (1.48 - 1.48i)T - 2iT^{2} \)
5 \( 1 + (-0.507 + 1.89i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.648 + 2.42i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 7.32T + 17T^{2} \)
19 \( 1 + (-0.930 + 0.249i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.63iT - 23T^{2} \)
29 \( 1 + (5.21 - 9.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.92 - 0.782i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.974 + 0.974i)T + 37iT^{2} \)
41 \( 1 + (-0.710 + 0.190i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.1 + 5.84i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.62 + 0.971i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.15 - 2.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.35 - 3.35i)T - 59iT^{2} \)
61 \( 1 + (8.18 + 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 1.62i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.32 - 2.23i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.01 - 7.50i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.31 + 4.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (2.89 - 2.89i)T - 89iT^{2} \)
97 \( 1 + (2.63 - 9.85i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.26731778606383381753009058318, −10.77982570040334654676350428515, −9.461283287351213855552592109147, −8.861080252224096199196967833848, −8.412408498002382488360923352111, −7.32265678279825373570775098856, −6.12725744537655281154649745972, −5.39976250833527160809420969508, −3.35129268707439589369376455763, −1.25977166285014158808856355809, 1.43638583890413308399223368606, 3.00825813205133769985816716783, 3.85256493952551295069246583419, 5.84622842652075533230537399771, 7.43560274374089837259203150727, 7.979038696618159978575473704279, 9.515951226119806797786478667007, 9.824428689561509738931444613754, 10.70737033756644705306337898857, 11.34566401069393290002825824073

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