Properties

Label 2-273-273.68-c1-0-15
Degree $2$
Conductor $273$
Sign $0.957 + 0.287i$
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 + 2·4-s + (2 + 1.73i)7-s + (1.5 + 2.59i)9-s + (−3 − 1.73i)12-s + (−1 − 3.46i)13-s + 4·16-s + (7.5 − 4.33i)19-s + (−1.50 − 4.33i)21-s + (2.5 + 4.33i)25-s − 5.19i·27-s + (4 + 3.46i)28-s + (−7.5 + 4.33i)31-s + (3 + 5.19i)36-s − 11·37-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + 4-s + (0.755 + 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.866 − 0.499i)12-s + (−0.277 − 0.960i)13-s + 16-s + (1.72 − 0.993i)19-s + (−0.327 − 0.944i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.755 + 0.654i)28-s + (−1.34 + 0.777i)31-s + (0.5 + 0.866i)36-s − 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.957 + 0.287i$
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: \((2,\ 273,\ (\ :1/2),\ 0.957 + 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30463 - 0.191328i\)
\(L(\frac12)\) \(\approx\) \(1.30463 - 0.191328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 - 2T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-7.5 + 4.33i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + (13.5 - 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.87147962314324408927532650214, −11.05492240262172909463277311833, −10.39319952387523364057149836898, −8.910111944363342481758103982556, −7.53654773184354997369529670453, −7.11526932273722673105108455125, −5.64101106934306220767374995501, −5.18736756205102008593806676804, −2.96282131225023266566178098366, −1.49903082663164853734163490938, 1.55479489252007004726710845250, 3.56940680954281740674240294168, 4.83483730757439600057775632918, 5.91618307430411832413316018983, 6.99803199730055645006653172379, 7.75752812144092765319546872157, 9.358892255116999374650434056434, 10.34030112497915912392582765464, 11.04352919388191984109753968112, 11.80201489812834484710536665064

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