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Properties

Label 2-546-91.4-c1-0-7
Degree $2$
Conductor $546$
Sign $0.848 + 0.529i$
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  − i·2-s + (−0.5 − 0.866i)3-s − 4-s + (−2.59 + 1.49i)5-s + (−0.866 + 0.5i)6-s + (2.50 + 0.844i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.59i)10-s + (−0.672 + 0.388i)11-s + (0.5 + 0.866i)12-s + (3.39 − 1.20i)13-s + (0.844 − 2.50i)14-s + (2.59 + 1.49i)15-s + 16-s + 2.34·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.288 − 0.499i)3-s − 0.5·4-s + (−1.16 + 0.670i)5-s + (−0.353 + 0.204i)6-s + (0.947 + 0.319i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.473 + 0.820i)10-s + (−0.202 + 0.117i)11-s + (0.144 + 0.249i)12-s + (0.942 − 0.334i)13-s + (0.225 − 0.670i)14-s + (0.670 + 0.386i)15-s + 0.250·16-s + 0.567·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.848 + 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09457 - 0.313712i\)
\(L(\frac12)\) \(\approx\) \(1.09457 - 0.313712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.50 - 0.844i)T \)
13 \( 1 + (-3.39 + 1.20i)T \)
good5 \( 1 + (2.59 - 1.49i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.672 - 0.388i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.34T + 17T^{2} \)
19 \( 1 + (-2.56 - 1.47i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.07T + 23T^{2} \)
29 \( 1 + (-0.541 + 0.937i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.31 - 3.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.87iT - 37T^{2} \)
41 \( 1 + (-9.81 - 5.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.64 + 4.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.35 - 4.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.30 + 3.99i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.79iT - 59T^{2} \)
61 \( 1 + (6.68 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.62 + 2.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.56 - 2.05i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.95 - 4.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.49 + 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.25iT - 83T^{2} \)
89 \( 1 + 8.71iT - 89T^{2} \)
97 \( 1 + (-2.45 + 1.41i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.01234976116003394672760504263, −10.20277345348201191696705035498, −8.809445216250942877475399221597, −7.967450895334229178998684614784, −7.42360084991720042681794965862, −6.07192945099829609432150152892, −4.97167142949154572544997894179, −3.81458871230196043048986973872, −2.75398747055717432157987869375, −1.16275267780309913903354548353, 0.926501790578839092399208060978, 3.50711904808966999857571842420, 4.47973540822008933316117022621, 5.07048248091692081249753649103, 6.26715394099131946777182774308, 7.50796048347138312474724931779, 8.159838941294453792458816079175, 8.826436020698364539523486359556, 9.914487076795650058882461032546, 11.08229026968842441619518011002

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