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Properties

Label 2-273-13.10-c1-0-11
Degree $2$
Conductor $273$
Sign $-0.709 + 0.705i$
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  + (−0.671 + 0.387i)2-s + (−0.5 − 0.866i)3-s + (−0.699 + 1.21i)4-s − 3.03i·5-s + (0.671 + 0.387i)6-s + (0.866 + 0.5i)7-s − 2.63i·8-s + (−0.499 + 0.866i)9-s + (1.17 + 2.03i)10-s + (−5.32 + 3.07i)11-s + 1.39·12-s + (−3.49 − 0.898i)13-s − 0.775·14-s + (−2.62 + 1.51i)15-s + (−0.375 − 0.650i)16-s + (3.22 − 5.58i)17-s + ⋯
L(s)  = 1  + (−0.475 + 0.274i)2-s + (−0.288 − 0.499i)3-s + (−0.349 + 0.605i)4-s − 1.35i·5-s + (0.274 + 0.158i)6-s + (0.327 + 0.188i)7-s − 0.932i·8-s + (−0.166 + 0.288i)9-s + (0.372 + 0.644i)10-s + (−1.60 + 0.926i)11-s + 0.403·12-s + (−0.968 − 0.249i)13-s − 0.207·14-s + (−0.678 + 0.391i)15-s + (−0.0938 − 0.162i)16-s + (0.781 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.709 + 0.705i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.709 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143685 - 0.348291i\)
\(L(\frac12)\) \(\approx\) \(0.143685 - 0.348291i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.49 + 0.898i)T \)
good2 \( 1 + (0.671 - 0.387i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.03iT - 5T^{2} \)
11 \( 1 + (5.32 - 3.07i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.74 + 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.46 + 7.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.00iT - 31T^{2} \)
37 \( 1 + (2.06 - 1.19i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.818 - 0.472i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.64 - 4.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.212iT - 47T^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (-1.49 - 0.861i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.76 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.50 + 3.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.12 + 3.53i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.75iT - 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 4.99iT - 83T^{2} \)
89 \( 1 + (3.08 - 1.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.36 + 0.787i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.98984496406847132128250886495, −10.36934971128543705757792521219, −9.500871483745070240441689044243, −8.384028022615877550531399363227, −7.889562100009733192876454059228, −6.91526163848017526333216574699, −5.08358373902423025778266139383, −4.73363666443893576662416056874, −2.49904813453131525438500628654, −0.33118687026746805255249679418, 2.27668666330297373085927859866, 3.76372334964805316717403444602, 5.34683910170157275906243404589, 6.04442005230203740465230900872, 7.59080800454821351305861193324, 8.419778226533379861566798076748, 9.952966097940066190611032219549, 10.31866742605387717185529888094, 10.92130999670800374476255118912, 11.83217246597071874042806691266

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