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Properties

Label 2-273-91.9-c1-0-16
Degree $2$
Conductor $273$
Sign $0.0657 + 0.997i$
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  + (−0.328 − 0.568i)2-s + 3-s + (0.784 − 1.35i)4-s + (−0.0109 + 0.0190i)5-s + (−0.328 − 0.568i)6-s + (−1.07 − 2.41i)7-s − 2.34·8-s + 9-s + 0.0144·10-s + 3.39·11-s + (0.784 − 1.35i)12-s + (−2.82 − 2.23i)13-s + (−1.02 + 1.40i)14-s + (−0.0109 + 0.0190i)15-s + (−0.799 − 1.38i)16-s + (−1.66 + 2.88i)17-s + ⋯
L(s)  = 1  + (−0.232 − 0.402i)2-s + 0.577·3-s + (0.392 − 0.679i)4-s + (−0.00490 + 0.00850i)5-s + (−0.134 − 0.232i)6-s + (−0.405 − 0.914i)7-s − 0.828·8-s + 0.333·9-s + 0.00455·10-s + 1.02·11-s + (0.226 − 0.392i)12-s + (−0.783 − 0.621i)13-s + (−0.273 + 0.375i)14-s + (−0.00283 + 0.00490i)15-s + (−0.199 − 0.346i)16-s + (−0.403 + 0.699i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0657 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0657 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0657 + 0.997i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.0657 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02254 - 0.957420i\)
\(L(\frac12)\) \(\approx\) \(1.02254 - 0.957420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (1.07 + 2.41i)T \)
13 \( 1 + (2.82 + 2.23i)T \)
good2 \( 1 + (0.328 + 0.568i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.0109 - 0.0190i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 3.39T + 11T^{2} \)
17 \( 1 + (1.66 - 2.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 3.33T + 19T^{2} \)
23 \( 1 + (-2.75 - 4.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.52 + 6.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.13 + 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.45 + 4.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.27 + 2.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.49 - 7.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.08 - 7.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.68 - 9.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.09 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + (3.57 + 6.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.102 - 0.176i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.42 - 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.71 - 8.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.47606599915238676833464382377, −10.67651342632304877102018500234, −9.680591428215765463520514237602, −9.239746675924476485639691902680, −7.68974970307324474032657577760, −6.86280388634312577382102418743, −5.69011451117674470626262677225, −4.10354839030039153625011441211, −2.85486246108600686817853845076, −1.22123858530204146913406869796, 2.37496741727909642019694104350, 3.42602916006640846648842200365, 4.98261222402315473245330135667, 6.68179508920965538372809889532, 7.00535212688924944979770977537, 8.570556326763595010182101054460, 8.896539530667126391370219810303, 9.925683250675842544585535586665, 11.46564598861291910147470371365, 12.17147606608278945248047888917

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