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Properties

Label 2-273-91.54-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.837 + 0.546i$
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  + (−0.926 + 0.926i)2-s + (−0.866 − 0.5i)3-s + 0.284i·4-s + (0.409 − 0.109i)5-s + (1.26 − 0.339i)6-s + (−2.25 + 1.38i)7-s + (−2.11 − 2.11i)8-s + (0.499 + 0.866i)9-s + (−0.277 + 0.481i)10-s + (1.35 − 0.362i)11-s + (0.142 − 0.246i)12-s + (−3.54 + 0.648i)13-s + (0.809 − 3.36i)14-s + (−0.409 − 0.109i)15-s + 3.35·16-s − 6.94·17-s + ⋯
L(s)  = 1  + (−0.654 + 0.654i)2-s + (−0.499 − 0.288i)3-s + 0.142i·4-s + (0.183 − 0.0491i)5-s + (0.516 − 0.138i)6-s + (−0.852 + 0.522i)7-s + (−0.748 − 0.748i)8-s + (0.166 + 0.288i)9-s + (−0.0879 + 0.152i)10-s + (0.408 − 0.109i)11-s + (0.0410 − 0.0710i)12-s + (−0.983 + 0.179i)13-s + (0.216 − 0.900i)14-s + (−0.105 − 0.0283i)15-s + 0.837·16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.837 + 0.546i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0338772 - 0.113930i\)
\(L(\frac12)\) \(\approx\) \(0.0338772 - 0.113930i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (2.25 - 1.38i)T \)
13 \( 1 + (3.54 - 0.648i)T \)
good2 \( 1 + (0.926 - 0.926i)T - 2iT^{2} \)
5 \( 1 + (-0.409 + 0.109i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.35 + 0.362i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 + (0.143 - 0.535i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.84iT - 23T^{2} \)
29 \( 1 + (3.01 + 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.17 - 8.09i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.24 + 4.24i)T + 37iT^{2} \)
41 \( 1 + (0.434 - 1.62i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.49 - 3.74i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.62 - 9.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.77 - 6.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.83 - 4.83i)T - 59iT^{2} \)
61 \( 1 + (-2.38 + 1.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.70 - 13.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.00 - 3.74i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.0 + 2.96i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.32 + 7.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.26 + 2.26i)T + 83iT^{2} \)
89 \( 1 + (8.19 - 8.19i)T - 89iT^{2} \)
97 \( 1 + (-15.5 + 4.17i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.50969043472050773091348697878, −11.65155311058072524173636759744, −10.39633364436860511407151719256, −9.296361100828393642267895250917, −8.765634197841786256347320137779, −7.42660193285505476196741768595, −6.64951790897965281179438159358, −5.89016913057774020104426201468, −4.27699702176717919539599459858, −2.54968190412563761200551463930, 0.11010877554594872591664982670, 2.11810540721708852282391318209, 3.76730644351384680354727743153, 5.22895577913175593467735382188, 6.32699424974944731841706198818, 7.35560056031069571032321052193, 8.992860764770341120829167098252, 9.617318366866256739660258800545, 10.31454177400243952916042084506, 11.19897223078956536887406887727

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