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Properties

Label 2-273-13.9-c1-0-5
Degree $2$
Conductor $273$
Sign $0.975 - 0.219i$
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  + (−0.100 + 0.174i)2-s + (0.5 − 0.866i)3-s + (0.979 + 1.69i)4-s + 3.75·5-s + (0.100 + 0.174i)6-s + (−0.5 − 0.866i)7-s − 0.798·8-s + (−0.499 − 0.866i)9-s + (−0.378 + 0.656i)10-s + (−2.87 + 4.98i)11-s + 1.95·12-s + (0.121 − 3.60i)13-s + 0.201·14-s + (1.87 − 3.25i)15-s + (−1.87 + 3.25i)16-s + (−2.10 − 3.63i)17-s + ⋯
L(s)  = 1  + (−0.0712 + 0.123i)2-s + (0.288 − 0.499i)3-s + (0.489 + 0.848i)4-s + 1.68·5-s + (0.0411 + 0.0712i)6-s + (−0.188 − 0.327i)7-s − 0.282·8-s + (−0.166 − 0.288i)9-s + (−0.119 + 0.207i)10-s + (−0.868 + 1.50i)11-s + 0.565·12-s + (0.0336 − 0.999i)13-s + 0.0538·14-s + (0.485 − 0.840i)15-s + (−0.469 + 0.813i)16-s + (−0.509 − 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.975 - 0.219i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.975 - 0.219i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71985 + 0.191501i\)
\(L(\frac12)\) \(\approx\) \(1.71985 + 0.191501i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.121 + 3.60i)T \)
good2 \( 1 + (0.100 - 0.174i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
11 \( 1 + (2.87 - 4.98i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.10 + 3.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.701 + 1.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.479 - 0.830i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.39 - 2.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 + (-5.03 + 8.72i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.40 - 2.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.479 + 0.830i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.67T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + (-6.19 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.66 + 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.80 - 6.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.81 + 3.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.95T + 73T^{2} \)
79 \( 1 - 3.56T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + (-0.903 + 1.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.13 - 14.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.25693816917133824036964017756, −10.87236172974145186836970623178, −9.946031847866548554727532002271, −9.142939216860416757985710821532, −7.84450326791948002192311571371, −7.09207242236394515126847741346, −6.15276237210248222872376886924, −4.87560986069220489586348275210, −2.89755575154883332849319403192, −2.06283378374686996872715104042, 1.82138220819548978445135977829, 2.90397292619631800623651374042, 4.91369729691846214089469724401, 6.04077659450855728669164355594, 6.33715071824402204123908696742, 8.345074085396179602791628698994, 9.257892850055399135233114592875, 10.01795762597318453968121501557, 10.68262254867341371540376722951, 11.54410052414760461298003477370

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