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Properties

Label 2-273-273.194-c1-0-17
Degree $2$
Conductor $273$
Sign $0.953 - 0.301i$
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  + (1.5 − 0.866i)3-s + (1 + 1.73i)4-s + (0.5 + 2.59i)7-s + (1.5 − 2.59i)9-s + (3 + 1.73i)12-s + (−3.5 + 0.866i)13-s + (−1.99 + 3.46i)16-s + (3.5 − 6.06i)19-s + (3 + 3.46i)21-s + (−2.5 − 4.33i)25-s − 5.19i·27-s + (−4 + 3.46i)28-s + (−3.5 − 6.06i)31-s + 6·36-s + (10.5 + 6.06i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.5 + 0.866i)4-s + (0.188 + 0.981i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.499i)12-s + (−0.970 + 0.240i)13-s + (−0.499 + 0.866i)16-s + (0.802 − 1.39i)19-s + (0.654 + 0.755i)21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (−0.755 + 0.654i)28-s + (−0.628 − 1.08i)31-s + 36-s + (1.72 + 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.953 - 0.301i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78228 + 0.274781i\)
\(L(\frac12)\) \(\approx\) \(1.78228 + 0.274781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
13 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-10.5 - 6.06i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−11.94646614137454396352629979764, −11.45295539594080733592142442171, −9.738880242290970064130700907986, −8.952072348664637585990870387160, −7.997008540921849858929910544936, −7.28343775246572175703810103926, −6.21393502990790940835217823366, −4.54147780566097017972001263234, −2.99803856267134099805939180767, −2.22688617377351818073348755434, 1.67886043423194948274307306892, 3.24985603416689115049053828135, 4.57239956789591955913348624517, 5.67650974607376112580475113096, 7.23276783045830538793381531072, 7.77395084408797116776956823434, 9.270818919308251311088690319125, 10.03361538651004428937512375350, 10.60668621886620810502034235683, 11.65496242618452171295814140877

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