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Properties

Label 2-273-91.59-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.0699 + 0.997i$
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  + (−0.507 − 0.507i)2-s + (0.866 − 0.5i)3-s − 1.48i·4-s + (1.10 + 0.295i)5-s + (−0.693 − 0.185i)6-s + (0.718 − 2.54i)7-s + (−1.76 + 1.76i)8-s + (0.499 − 0.866i)9-s + (−0.409 − 0.709i)10-s + (1.72 + 0.462i)11-s + (−0.741 − 1.28i)12-s + (−3.60 + 0.189i)13-s + (−1.65 + 0.928i)14-s + (1.10 − 0.295i)15-s − 1.16·16-s + 3.69·17-s + ⋯
L(s)  = 1  + (−0.359 − 0.359i)2-s + (0.499 − 0.288i)3-s − 0.741i·4-s + (0.492 + 0.131i)5-s + (−0.283 − 0.0759i)6-s + (0.271 − 0.962i)7-s + (−0.625 + 0.625i)8-s + (0.166 − 0.288i)9-s + (−0.129 − 0.224i)10-s + (0.520 + 0.139i)11-s + (−0.214 − 0.370i)12-s + (−0.998 + 0.0525i)13-s + (−0.443 + 0.248i)14-s + (0.284 − 0.0761i)15-s − 0.292·16-s + 0.895·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0699 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0699 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0699 + 0.997i$
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: \((2,\ 273,\ (\ :1/2),\ -0.0699 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894575 - 0.959497i\)
\(L(\frac12)\) \(\approx\) \(0.894575 - 0.959497i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.718 + 2.54i)T \)
13 \( 1 + (3.60 - 0.189i)T \)
good2 \( 1 + (0.507 + 0.507i)T + 2iT^{2} \)
5 \( 1 + (-1.10 - 0.295i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.72 - 0.462i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
19 \( 1 + (-0.160 - 0.600i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 + (0.0743 - 0.128i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.24 - 4.64i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.17 + 3.17i)T - 37iT^{2} \)
41 \( 1 + (-1.12 - 4.19i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.81 + 3.93i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0779 - 0.291i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.0 - 10.0i)T + 59iT^{2} \)
61 \( 1 + (-2.51 - 1.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.12 - 11.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.67 - 13.6i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11.3 + 3.05i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.537 + 0.930i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.90 + 3.90i)T - 83iT^{2} \)
89 \( 1 + (8.17 + 8.17i)T + 89iT^{2} \)
97 \( 1 + (-4.98 - 1.33i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.56478953468947564264401151577, −10.39506249756449448773686528814, −9.923297670403308980558986877399, −9.014139508389165807144350933523, −7.78224581525276544378710140571, −6.81026552035240578965395980975, −5.63884739060791536275782932090, −4.28105424496889933686373753917, −2.56619837157546278476575119970, −1.21348801148057270915944813068, 2.30829772882960970147189735221, 3.56853858088592192350894638880, 5.05567612630377491142481956314, 6.24833139199074422445730510702, 7.58027451182806739100360068641, 8.259113669776991299081295430087, 9.422425352482348622050193009925, 9.641248690105021559680666806309, 11.36357572390073140247754107357, 12.18432850883324086397837801741

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