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Properties

Label 2-546-91.74-c1-0-14
Degree $2$
Conductor $546$
Sign $0.193 + 0.981i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.924 − 1.60i)5-s + (−0.5 − 0.866i)6-s + (2.61 − 0.405i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.924 − 1.60i)10-s + (−0.357 − 0.619i)11-s + (−0.5 − 0.866i)12-s + (−2.81 − 2.25i)13-s + (2.61 − 0.405i)14-s + (−0.924 + 1.60i)15-s + 16-s + 4.31·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.413 − 0.716i)5-s + (−0.204 − 0.353i)6-s + (0.988 − 0.153i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.292 − 0.506i)10-s + (−0.107 − 0.186i)11-s + (−0.144 − 0.249i)12-s + (−0.780 − 0.625i)13-s + (0.698 − 0.108i)14-s + (−0.238 + 0.413i)15-s + 0.250·16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.193 + 0.981i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52308 - 1.25232i\)
\(L(\frac12)\) \(\approx\) \(1.52308 - 1.25232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.61 + 0.405i)T \)
13 \( 1 + (2.81 + 2.25i)T \)
good5 \( 1 + (0.924 + 1.60i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.357 + 0.619i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.31T + 17T^{2} \)
19 \( 1 + (-3.43 + 5.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.16T + 23T^{2} \)
29 \( 1 + (4.63 - 8.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.11 + 1.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.13T + 37T^{2} \)
41 \( 1 + (-2.23 + 3.86i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0979 + 0.169i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.60 - 7.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.80 + 6.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.48T + 59T^{2} \)
61 \( 1 + (4.31 - 7.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.96 - 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.50 - 13.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.989 - 1.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.37 - 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.71T + 83T^{2} \)
89 \( 1 + 4.46T + 89T^{2} \)
97 \( 1 + (-1.39 - 2.41i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.94269967068020183342607338317, −9.871856967744027588879618805611, −8.559285144052906372925095333840, −7.74082788201413640773906455042, −7.13281071947339412944742668543, −5.61385594766748840152352294320, −5.11955948782836035251700652734, −4.07405097994514655746087476318, −2.58099667236750177719285108712, −1.02161635613942835727665879266, 1.99558881640034927576938225985, 3.44463184114609447712093092399, 4.33017157196192280283613355380, 5.35183455530185887036314987674, 6.17229875817698034695796918322, 7.54621525518485251568066060239, 7.908603714537607633102082334055, 9.531742474478206276403384848914, 10.26252973919443612862649330878, 11.14724474417880121820074799080

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