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Properties

Label 2-273-91.4-c1-0-9
Degree $2$
Conductor $273$
Sign $0.994 - 0.100i$
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  − 0.560i·2-s + (0.5 + 0.866i)3-s + 1.68·4-s + (0.449 − 0.259i)5-s + (0.485 − 0.280i)6-s + (0.680 + 2.55i)7-s − 2.06i·8-s + (−0.499 + 0.866i)9-s + (−0.145 − 0.252i)10-s + (−1.87 + 1.08i)11-s + (0.842 + 1.45i)12-s + (3.29 − 1.45i)13-s + (1.43 − 0.381i)14-s + (0.449 + 0.259i)15-s + 2.21·16-s − 4.96·17-s + ⋯
L(s)  = 1  − 0.396i·2-s + (0.288 + 0.499i)3-s + 0.842·4-s + (0.201 − 0.116i)5-s + (0.198 − 0.114i)6-s + (0.257 + 0.966i)7-s − 0.730i·8-s + (−0.166 + 0.288i)9-s + (−0.0460 − 0.0798i)10-s + (−0.565 + 0.326i)11-s + (0.243 + 0.421i)12-s + (0.915 − 0.403i)13-s + (0.383 − 0.101i)14-s + (0.116 + 0.0670i)15-s + 0.552·16-s − 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70346 + 0.0861276i\)
\(L(\frac12)\) \(\approx\) \(1.70346 + 0.0861276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.680 - 2.55i)T \)
13 \( 1 + (-3.29 + 1.45i)T \)
good2 \( 1 + 0.560iT - 2T^{2} \)
5 \( 1 + (-0.449 + 0.259i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.87 - 1.08i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + (4.78 + 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.79T + 23T^{2} \)
29 \( 1 + (-3.81 + 6.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.15 - 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.84iT - 37T^{2} \)
41 \( 1 + (9.23 + 5.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.11 + 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.69 - 5.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.04 + 1.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.04iT - 59T^{2} \)
61 \( 1 + (3.12 - 5.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.71 - 0.990i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.61 - 4.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.47 + 1.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.48 - 4.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.65iT - 83T^{2} \)
89 \( 1 + 2.53iT - 89T^{2} \)
97 \( 1 + (-8.01 + 4.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.68755870724473051476562157969, −10.99207298271887253333549842881, −10.22963656539207068903094768449, −9.033945158119784063151053640058, −8.320486373752580971577984400789, −6.91558167634563872415415128267, −5.86122878106662860816576438293, −4.63312317640512246639440457187, −3.05998827576208887884968242218, −2.05169990449381467314663423366, 1.67375525259651741444878934390, 3.13487473926262216543962393731, 4.74274133960538376839032581562, 6.44250134709325089780982636372, 6.68412048645593139693436274192, 8.018249223969919742039195107130, 8.580632053404907172529694579153, 10.25799852825873004184276097673, 10.94582382080014142886241828667, 11.73750330561962326247383067026

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