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Properties

Label 2-273-7.2-c1-0-9
Degree $2$
Conductor $273$
Sign $0.818 - 0.574i$
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  + (−0.0388 + 0.0672i)2-s + (0.5 + 0.866i)3-s + (0.996 + 1.72i)4-s + (1.10 − 1.90i)5-s − 0.0776·6-s + (2.64 − 0.0672i)7-s − 0.310·8-s + (−0.499 + 0.866i)9-s + (0.0855 + 0.148i)10-s + (−1.45 − 2.52i)11-s + (−0.996 + 1.72i)12-s − 13-s + (−0.0981 + 0.180i)14-s + 2.20·15-s + (−1.98 + 3.43i)16-s + (0.196 + 0.341i)17-s + ⋯
L(s)  = 1  + (−0.0274 + 0.0475i)2-s + (0.288 + 0.499i)3-s + (0.498 + 0.863i)4-s + (0.492 − 0.852i)5-s − 0.0317·6-s + (0.999 − 0.0254i)7-s − 0.109·8-s + (−0.166 + 0.288i)9-s + (0.0270 + 0.0468i)10-s + (−0.439 − 0.761i)11-s + (−0.287 + 0.498i)12-s − 0.277·13-s + (−0.0262 + 0.0482i)14-s + 0.568·15-s + (−0.495 + 0.858i)16-s + (0.0477 + 0.0827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60017 + 0.505884i\)
\(L(\frac12)\) \(\approx\) \(1.60017 + 0.505884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.64 + 0.0672i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.0388 - 0.0672i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.10 + 1.90i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.45 + 2.52i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.196 - 0.341i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0623 - 0.108i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.69 - 2.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.642T + 29T^{2} \)
31 \( 1 + (2.97 + 5.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.09 - 3.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.434T + 41T^{2} \)
43 \( 1 + 6.74T + 43T^{2} \)
47 \( 1 + (-4.73 + 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.21 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.66 - 6.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.19 + 2.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.07 + 5.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + (-3.77 - 6.54i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.18 + 2.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.78T + 83T^{2} \)
89 \( 1 + (-3.20 + 5.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−11.85034844066598553038672625726, −11.20186547737496851652541002139, −10.08780784060744257945004004463, −8.871655961571537012566422768189, −8.286130296011721206751744728124, −7.37865487824049865355116545737, −5.77114270021148784098211248784, −4.78739935785386298098803501587, −3.49858640358184053307376660369, −1.98845905607278322101647626027, 1.73499768391214493231037139253, 2.67949049897543853959360811084, 4.76694391015261845560203288279, 5.90193486282536522176264429331, 6.92539499247113263524569323744, 7.67381886000814504562417125102, 9.013790963004416974365951101609, 10.19025855057495809140471430863, 10.71777447904852497400494264103, 11.70778431416009833719716714127

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