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Properties

Label 2-546-21.17-c1-0-27
Degree $2$
Conductor $546$
Sign $-0.856 - 0.516i$
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  + (−0.866 + 0.5i)2-s + (0.115 − 1.72i)3-s + (0.499 − 0.866i)4-s + (−1.85 − 3.22i)5-s + (0.763 + 1.55i)6-s + (−1.49 + 2.17i)7-s + 0.999i·8-s + (−2.97 − 0.400i)9-s + (3.22 + 1.85i)10-s + (0.553 + 0.319i)11-s + (−1.43 − 0.964i)12-s − i·13-s + (0.209 − 2.63i)14-s + (−5.78 + 2.84i)15-s + (−0.5 − 0.866i)16-s + (3.63 − 6.30i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.0668 − 0.997i)3-s + (0.249 − 0.433i)4-s + (−0.831 − 1.44i)5-s + (0.311 + 0.634i)6-s + (−0.566 + 0.823i)7-s + 0.353i·8-s + (−0.991 − 0.133i)9-s + (1.01 + 0.588i)10-s + (0.166 + 0.0963i)11-s + (−0.415 − 0.278i)12-s − 0.277i·13-s + (0.0558 − 0.704i)14-s + (−1.49 + 0.733i)15-s + (−0.125 − 0.216i)16-s + (0.882 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.856 - 0.516i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0583289 + 0.209603i\)
\(L(\frac12)\) \(\approx\) \(0.0583289 + 0.209603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.115 + 1.72i)T \)
7 \( 1 + (1.49 - 2.17i)T \)
13 \( 1 + iT \)
good5 \( 1 + (1.85 + 3.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.553 - 0.319i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.63 + 6.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.62 - 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.07 - 2.35i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.98iT - 29T^{2} \)
31 \( 1 + (-0.608 - 0.351i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.21 - 9.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 3.57T + 43T^{2} \)
47 \( 1 + (-3.53 - 6.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.65 + 3.84i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.35 + 2.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.0 + 5.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.15 - 1.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (8.28 + 4.78i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.00 + 8.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (1.95 + 3.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.50iT - 97T^{2} \)
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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

−9.943837345460011532130236173814, −9.130414056284429287212812737403, −8.328940807906066326721376575259, −7.893983646378808178116869309401, −6.78606475559296099735622965524, −5.77461196918116660263307098196, −4.88921882190755283786074968145, −3.18452064570368490556914400744, −1.56995582560048583903038969281, −0.15067927709302213717046061886, 2.56659605563159690048308660390, 3.75722957363745200086704299847, 4.04675522451594690613173950769, 6.07949276650405003782036272126, 6.89288276063621233180199718610, 7.912308122731710998783572413680, 8.685642473027394734699989936093, 10.10516954995123236468935975285, 10.23012286815916011809436489735, 11.04674715052541437775358810351

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