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Properties

Label 2-273-91.81-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.521 - 0.853i$
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  + (−0.532 + 0.922i)2-s − 3-s + (0.432 + 0.748i)4-s + (1.19 + 2.06i)5-s + (0.532 − 0.922i)6-s + (2.58 − 0.554i)7-s − 3.05·8-s + 9-s − 2.53·10-s − 0.666·11-s + (−0.432 − 0.748i)12-s + (2.19 + 2.85i)13-s + (−0.866 + 2.68i)14-s + (−1.19 − 2.06i)15-s + (0.761 − 1.31i)16-s + (−0.707 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.376 + 0.652i)2-s − 0.577·3-s + (0.216 + 0.374i)4-s + (0.532 + 0.921i)5-s + (0.217 − 0.376i)6-s + (0.977 − 0.209i)7-s − 1.07·8-s + 0.333·9-s − 0.802·10-s − 0.200·11-s + (−0.124 − 0.216i)12-s + (0.609 + 0.792i)13-s + (−0.231 + 0.716i)14-s + (−0.307 − 0.532i)15-s + (0.190 − 0.329i)16-s + (−0.171 − 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506080 + 0.902075i\)
\(L(\frac12)\) \(\approx\) \(0.506080 + 0.902075i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.554i)T \)
13 \( 1 + (-2.19 - 2.85i)T \)
good2 \( 1 + (0.532 - 0.922i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.19 - 2.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.666T + 11T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 + (2.99 - 5.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.647 - 1.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.94 + 6.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.26 - 9.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.54 + 9.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.39 + 5.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.57 - 4.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.83T + 61T^{2} \)
67 \( 1 - 5.57T + 67T^{2} \)
71 \( 1 + (-6.01 + 10.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.05 - 7.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.00 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.44T + 83T^{2} \)
89 \( 1 + (0.910 - 1.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.88 - 13.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.94631053050131076428984175448, −11.22673952206282167115850608091, −10.50357055605003446731261121259, −9.240211192230253255529407803318, −8.190602715077783893195316863274, −7.17501148677285708555739593200, −6.49006094180849711603505799003, −5.46095754003224818442256069772, −3.88246207346548257731306459917, −2.15876622638366465719306284618, 1.02151521077452764502297709509, 2.28358400564231869592980449805, 4.43373340963509898627950591255, 5.56486251624172481423103853416, 6.23053912686147654208178867935, 8.005385647955474952553350982609, 8.853577132251284810510428961580, 9.865179391188357060203868976354, 10.79953417455748070827384200501, 11.31593326097892123209707577001

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