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Properties

Label 2-273-91.80-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.998 + 0.0492i$
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  + (−0.478 − 1.78i)2-s − i·3-s + (−1.23 + 0.710i)4-s + (−0.0199 + 0.0744i)5-s + (−1.78 + 0.478i)6-s + (1.85 − 1.88i)7-s + (−0.758 − 0.758i)8-s − 9-s + 0.142·10-s + (−0.492 − 0.492i)11-s + (0.710 + 1.23i)12-s + (−2.39 − 2.69i)13-s + (−4.25 − 2.42i)14-s + (0.0744 + 0.0199i)15-s + (−2.41 + 4.17i)16-s + (−0.618 − 1.07i)17-s + ⋯
L(s)  = 1  + (−0.338 − 1.26i)2-s − 0.577i·3-s + (−0.615 + 0.355i)4-s + (−0.00892 + 0.0333i)5-s + (−0.729 + 0.195i)6-s + (0.702 − 0.711i)7-s + (−0.268 − 0.268i)8-s − 0.333·9-s + 0.0451·10-s + (−0.148 − 0.148i)11-s + (0.205 + 0.355i)12-s + (−0.665 − 0.746i)13-s + (−1.13 − 0.647i)14-s + (0.0192 + 0.00515i)15-s + (−0.602 + 1.04i)16-s + (−0.150 − 0.259i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0492i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.998 + 0.0492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0251844 - 1.02176i\)
\(L(\frac12)\) \(\approx\) \(0.0251844 - 1.02176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-1.85 + 1.88i)T \)
13 \( 1 + (2.39 + 2.69i)T \)
good2 \( 1 + (0.478 + 1.78i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.0199 - 0.0744i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.492 + 0.492i)T + 11iT^{2} \)
17 \( 1 + (0.618 + 1.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.98 - 3.98i)T + 19iT^{2} \)
23 \( 1 + (6.38 + 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.84 - 3.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.76 + 2.34i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.44 - 1.19i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.231 + 0.865i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.14 + 1.24i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.404 - 0.108i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.75 + 9.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.5 - 3.62i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 6.50iT - 61T^{2} \)
67 \( 1 + (1.47 - 1.47i)T - 67iT^{2} \)
71 \( 1 + (-0.119 - 0.445i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.31 + 4.91i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.26 - 5.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.54 - 9.54i)T + 83iT^{2} \)
89 \( 1 + (1.15 + 4.29i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-15.6 + 4.19i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.56362229484363815279693469980, −10.40967121653699985541210823771, −10.03624178824355309642574739214, −8.586014790815452559363472398972, −7.74017307579168615442816792080, −6.59206582254585573352137791516, −5.10433748491294839084721698594, −3.60426794760517190253090798176, −2.32309390599857630084759728526, −0.907096351311119534596488448199, 2.56022604618427385933695297987, 4.55803174258883939573557980176, 5.38509658774945869004766994786, 6.47403500365723851572586597683, 7.55826462724958314249853654592, 8.466493737771234406063518233855, 9.205375778203364996580172147285, 10.19244415247237225912878507538, 11.63090743716591426111120485858, 12.00450221369092266076958976595

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