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Properties

Label 2-273-91.59-c1-0-3
Degree $2$
Conductor $273$
Sign $0.230 + 0.973i$
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  + (−1.65 − 1.65i)2-s + (0.866 − 0.5i)3-s + 3.46i·4-s + (2.69 + 0.721i)5-s + (−2.25 − 0.604i)6-s + (−2.54 + 0.727i)7-s + (2.41 − 2.41i)8-s + (0.499 − 0.866i)9-s + (−3.25 − 5.64i)10-s + (5.85 + 1.56i)11-s + (1.73 + 2.99i)12-s + (0.711 − 3.53i)13-s + (5.40 + 3.00i)14-s + (2.69 − 0.721i)15-s − 1.06·16-s + 1.28·17-s + ⋯
L(s)  = 1  + (−1.16 − 1.16i)2-s + (0.499 − 0.288i)3-s + 1.73i·4-s + (1.20 + 0.322i)5-s + (−0.921 − 0.246i)6-s + (−0.961 + 0.274i)7-s + (0.854 − 0.854i)8-s + (0.166 − 0.288i)9-s + (−1.03 − 1.78i)10-s + (1.76 + 0.472i)11-s + (0.499 + 0.865i)12-s + (0.197 − 0.980i)13-s + (1.44 + 0.802i)14-s + (0.695 − 0.186i)15-s − 0.265·16-s + 0.311·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777103 - 0.614632i\)
\(L(\frac12)\) \(\approx\) \(0.777103 - 0.614632i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 0.727i)T \)
13 \( 1 + (-0.711 + 3.53i)T \)
good2 \( 1 + (1.65 + 1.65i)T + 2iT^{2} \)
5 \( 1 + (-2.69 - 0.721i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-5.85 - 1.56i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
19 \( 1 + (-0.502 - 1.87i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.14iT - 23T^{2} \)
29 \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.515 + 1.92i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.34 - 8.34i)T - 37iT^{2} \)
41 \( 1 + (1.19 + 4.44i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.94 - 1.70i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.67 - 9.99i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.30 + 7.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.22 - 8.22i)T + 59iT^{2} \)
61 \( 1 + (3.17 + 1.83i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.441 + 1.64i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.73 - 6.47i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.37 - 1.17i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.418 - 0.724i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.01 + 2.01i)T - 83iT^{2} \)
89 \( 1 + (3.52 + 3.52i)T + 89iT^{2} \)
97 \( 1 + (-6.27 - 1.68i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.70707419442244330689863361341, −10.25497972952138575219717393399, −9.897183786774854289365967020895, −9.162282779835342523031261100975, −8.315109240687953567748355048198, −6.88538638320227876707642245106, −5.96086932353503068962708992999, −3.62611639389516780878820523068, −2.59751345317245730706100416530, −1.39522748173177526883638613074, 1.47639906976281088814102820321, 3.67310983266850947537086685829, 5.47670507835892681106218817808, 6.53018981316879920902046628124, 7.00273008962998568892293835728, 8.639487210822363078747437426813, 9.198110728781746098586901307144, 9.613224211272395848559275097010, 10.57210850721051413596640751782, 12.07732758584613286647826006233

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