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Properties

Label 2-273-273.68-c1-0-14
Degree $2$
Conductor $273$
Sign $0.852 - 0.522i$
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  − 0.883i·2-s + (1.41 + 1.00i)3-s + 1.21·4-s + (−1.72 + 2.98i)5-s + (0.886 − 1.24i)6-s + (−2.26 + 1.36i)7-s − 2.84i·8-s + (0.987 + 2.83i)9-s + (2.63 + 1.52i)10-s + (3.32 + 1.92i)11-s + (1.72 + 1.22i)12-s + (0.144 − 3.60i)13-s + (1.20 + 2.00i)14-s + (−5.42 + 2.48i)15-s − 0.0734·16-s − 0.161·17-s + ⋯
L(s)  = 1  − 0.624i·2-s + (0.815 + 0.579i)3-s + 0.609·4-s + (−0.769 + 1.33i)5-s + (0.361 − 0.509i)6-s + (−0.856 + 0.516i)7-s − 1.00i·8-s + (0.329 + 0.944i)9-s + (0.832 + 0.480i)10-s + (1.00 + 0.579i)11-s + (0.497 + 0.353i)12-s + (0.0399 − 0.999i)13-s + (0.322 + 0.535i)14-s + (−1.39 + 0.641i)15-s − 0.0183·16-s − 0.0392·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.852 - 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57730 + 0.444748i\)
\(L(\frac12)\) \(\approx\) \(1.57730 + 0.444748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 - 1.00i)T \)
7 \( 1 + (2.26 - 1.36i)T \)
13 \( 1 + (-0.144 + 3.60i)T \)
good2 \( 1 + 0.883iT - 2T^{2} \)
5 \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.32 - 1.92i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.161T + 17T^{2} \)
19 \( 1 + (1.41 - 0.817i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.54iT - 23T^{2} \)
29 \( 1 + (-8.21 + 4.74i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.139 + 0.0806i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.98T + 37T^{2} \)
41 \( 1 + (-2.41 - 4.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.32 + 7.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.42 + 5.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.2 - 6.50i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.91T + 59T^{2} \)
61 \( 1 + (5.63 - 3.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.472 + 0.818i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.33 - 2.50i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.62 + 4.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.23 - 9.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.04T + 83T^{2} \)
89 \( 1 + 6.88T + 89T^{2} \)
97 \( 1 + (-6.97 - 4.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.97743036404141126016495096119, −10.77785804297190582524187760742, −10.33107305320177226875053686368, −9.423338803879536353428904056153, −8.129975303025906886879630483473, −7.06521897423686633484373669641, −6.27938134324141824758162851485, −4.12977862136574192770521273107, −3.19505717584090717201944133979, −2.48594619313901392692605762682, 1.35041247423016498322690075047, 3.30186867773944583011621658561, 4.46451383302327247903808552826, 6.20519794071398184326750957471, 6.94659627377530705252997528038, 7.896197555999826853038190045633, 8.782175783659580881947773381152, 9.413154000188210881996775592154, 11.12078601239097107040931340407, 12.09580742939775839416049785840

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