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Properties

Label 2-273-91.81-c1-0-13
Degree $2$
Conductor $273$
Sign $0.0787 + 0.996i$
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  + (0.415 − 0.719i)2-s − 3-s + (0.654 + 1.13i)4-s + (−1.30 − 2.26i)5-s + (−0.415 + 0.719i)6-s + (0.801 − 2.52i)7-s + 2.75·8-s + 9-s − 2.17·10-s − 1.84·11-s + (−0.654 − 1.13i)12-s + (2.74 − 2.33i)13-s + (−1.48 − 1.62i)14-s + (1.30 + 2.26i)15-s + (−0.165 + 0.287i)16-s + (−3.41 − 5.91i)17-s + ⋯
L(s)  = 1  + (0.293 − 0.509i)2-s − 0.577·3-s + (0.327 + 0.566i)4-s + (−0.585 − 1.01i)5-s + (−0.169 + 0.293i)6-s + (0.302 − 0.952i)7-s + 0.972·8-s + 0.333·9-s − 0.687·10-s − 0.557·11-s + (−0.188 − 0.327i)12-s + (0.761 − 0.648i)13-s + (−0.396 − 0.434i)14-s + (0.337 + 0.585i)15-s + (−0.0414 + 0.0717i)16-s + (−0.828 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0787 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0787 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0787 + 0.996i$
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: \((2,\ 273,\ (\ :1/2),\ 0.0787 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928279 - 0.857835i\)
\(L(\frac12)\) \(\approx\) \(0.928279 - 0.857835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.801 + 2.52i)T \)
13 \( 1 + (-2.74 + 2.33i)T \)
good2 \( 1 + (-0.415 + 0.719i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.30 + 2.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.84T + 11T^{2} \)
17 \( 1 + (3.41 + 5.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 5.06T + 19T^{2} \)
23 \( 1 + (-0.636 + 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.724 - 1.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.93 - 6.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.41 - 7.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.109 + 0.189i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.624 - 1.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.33 + 2.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.01 - 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.72T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + (-1.78 + 3.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.26 - 5.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.08 - 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.67T + 83T^{2} \)
89 \( 1 + (-7.57 + 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.08 + 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.60572687922893535389983356558, −11.09100084065164395124388799118, −10.11360909904013588262182861899, −8.648404122710574425112825469367, −7.72617217327945348355060435254, −6.93275302898951254184473545840, −5.13842886846279558722537875597, −4.43533470608291241187273205665, −3.16984446195345187470157784719, −1.04076859469616768000342724743, 2.07068356920162146572268786850, 3.88966561781571630848058281695, 5.33063893033043474680181767974, 6.10320729395386811749883916447, 6.99264324435405965257886498116, 7.952490658219753473300927915704, 9.312924999683606160079488445427, 10.70783591092713257536574055535, 11.03348976931531380781549270077, 11.88813707344972820683246199294

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