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Properties

Label 2-273-273.68-c1-0-1
Degree $2$
Conductor $273$
Sign $0.580 - 0.814i$
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  − 1.92i·2-s + (−1.08 − 1.35i)3-s − 1.70·4-s + (−1.94 + 3.37i)5-s + (−2.59 + 2.08i)6-s + (−0.630 + 2.56i)7-s − 0.575i·8-s + (−0.650 + 2.92i)9-s + (6.48 + 3.74i)10-s + (−2.13 − 1.23i)11-s + (1.84 + 2.29i)12-s + (−0.534 + 3.56i)13-s + (4.94 + 1.21i)14-s + (6.66 − 1.02i)15-s − 4.50·16-s − 6.77·17-s + ⋯
L(s)  = 1  − 1.36i·2-s + (−0.625 − 0.779i)3-s − 0.850·4-s + (−0.870 + 1.50i)5-s + (−1.06 + 0.851i)6-s + (−0.238 + 0.971i)7-s − 0.203i·8-s + (−0.216 + 0.976i)9-s + (2.05 + 1.18i)10-s + (−0.643 − 0.371i)11-s + (0.532 + 0.663i)12-s + (−0.148 + 0.988i)13-s + (1.32 + 0.324i)14-s + (1.72 − 0.264i)15-s − 1.12·16-s − 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.580 - 0.814i$
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: \((2,\ 273,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262795 + 0.135454i\)
\(L(\frac12)\) \(\approx\) \(0.262795 + 0.135454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 + 1.35i)T \)
7 \( 1 + (0.630 - 2.56i)T \)
13 \( 1 + (0.534 - 3.56i)T \)
good2 \( 1 + 1.92iT - 2T^{2} \)
5 \( 1 + (1.94 - 3.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.13 + 1.23i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.77T + 17T^{2} \)
19 \( 1 + (-2.32 + 1.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.78iT - 23T^{2} \)
29 \( 1 + (0.0349 - 0.0201i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.270 - 0.156i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.07T + 37T^{2} \)
41 \( 1 + (-1.73 - 3.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.35 - 9.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.67 - 4.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.37 - 2.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + (-3.28 + 1.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.23 + 5.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.84 - 5.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.25 - 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.62 + 4.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.83T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (12.5 + 7.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.64146110100949427358165047657, −11.32682726954372353612952523460, −10.73926569350566422037010976861, −9.521908766422932152855208256890, −8.187303746796787574234685719128, −6.91565162320140290069473034470, −6.34853626522261845055738808828, −4.54075194256555941094287527950, −2.97350964671531724831740128637, −2.26292751430126896556139996538, 0.22993927264052567240882577684, 3.92307479015978567137284403648, 4.84492125810939700881801178685, 5.47795439082415420166689603327, 6.86653044853539532853725256662, 7.81468024261303809533538629361, 8.630683355500318034111206254722, 9.643997485422930646977836323657, 10.81059420942519735284652667783, 11.73887375239082584805505509346

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