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Properties

Label 2-273-91.51-c1-0-0
Degree $2$
Conductor $273$
Sign $0.550 - 0.834i$
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  + (−1.34 − 0.779i)2-s + (−0.5 − 0.866i)3-s + (0.214 + 0.371i)4-s + (1.85 + 1.06i)5-s + 1.55i·6-s + (−1.34 + 2.27i)7-s + 2.44i·8-s + (−0.499 + 0.866i)9-s + (−1.66 − 2.88i)10-s + (−4.76 + 2.75i)11-s + (0.214 − 0.371i)12-s + (−2.90 − 2.13i)13-s + (3.59 − 2.01i)14-s − 2.13i·15-s + (2.33 − 4.04i)16-s + (3.71 + 6.43i)17-s + ⋯
L(s)  = 1  + (−0.954 − 0.550i)2-s + (−0.288 − 0.499i)3-s + (0.107 + 0.185i)4-s + (0.828 + 0.478i)5-s + 0.636i·6-s + (−0.510 + 0.860i)7-s + 0.865i·8-s + (−0.166 + 0.288i)9-s + (−0.526 − 0.912i)10-s + (−1.43 + 0.829i)11-s + (0.0618 − 0.107i)12-s + (−0.805 − 0.592i)13-s + (0.960 − 0.539i)14-s − 0.552i·15-s + (0.584 − 1.01i)16-s + (0.901 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420114 + 0.226160i\)
\(L(\frac12)\) \(\approx\) \(0.420114 + 0.226160i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 2.27i)T \)
13 \( 1 + (2.90 + 2.13i)T \)
good2 \( 1 + (1.34 + 0.779i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.85 - 1.06i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.76 - 2.75i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.71 - 6.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.20 - 1.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 3.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.903T + 29T^{2} \)
31 \( 1 + (2.62 - 1.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.824 - 0.476i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.86iT - 41T^{2} \)
43 \( 1 - 4.89T + 43T^{2} \)
47 \( 1 + (4.55 + 2.63i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.49 - 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.30 - 4.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.81 + 3.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.52 + 0.882i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (6.43 - 3.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.64 - 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 + (-4.49 - 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.06iT - 97T^{2} \)
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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

−12.14897979633878151349771394481, −10.70043840814928648348377237036, −10.13527573293344601139826386778, −9.565578964645926131998694391576, −8.225377220884056838195963567028, −7.45385491658947680862062677957, −5.84098474923943294913935586525, −5.39654242686156469760534572152, −2.80024317378229077749733152029, −1.87352524215023467071224723268, 0.49911874940107730086682852944, 3.08404091836720946810223505640, 4.73955631693382023261287971949, 5.77382796389421114962815057365, 7.06735126365894563986571104936, 7.84282824697704933946596008158, 9.106208921835673975306265521256, 9.780662250711983585659732637356, 10.28411976285234168489439478273, 11.55279001352017616495019267936

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