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Properties

Label 2-546-91.30-c1-0-3
Degree $2$
Conductor $546$
Sign $0.102 - 0.994i$
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 + 3-s + (0.499 − 0.866i)4-s + (1.54 + 0.889i)5-s + (−0.866 + 0.5i)6-s + (−0.542 + 2.58i)7-s + 0.999i·8-s + 9-s − 1.77·10-s − 0.0292i·11-s + (0.499 − 0.866i)12-s + (0.100 + 3.60i)13-s + (−0.824 − 2.51i)14-s + (1.54 + 0.889i)15-s + (−0.5 − 0.866i)16-s + (−3.05 + 5.29i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 0.577·3-s + (0.249 − 0.433i)4-s + (0.689 + 0.397i)5-s + (−0.353 + 0.204i)6-s + (−0.205 + 0.978i)7-s + 0.353i·8-s + 0.333·9-s − 0.562·10-s − 0.00881i·11-s + (0.144 − 0.249i)12-s + (0.0279 + 0.999i)13-s + (−0.220 − 0.671i)14-s + (0.397 + 0.229i)15-s + (−0.125 − 0.216i)16-s + (−0.741 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02445 + 0.924598i\)
\(L(\frac12)\) \(\approx\) \(1.02445 + 0.924598i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (0.542 - 2.58i)T \)
13 \( 1 + (-0.100 - 3.60i)T \)
good5 \( 1 + (-1.54 - 0.889i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.0292iT - 11T^{2} \)
17 \( 1 + (3.05 - 5.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 6.65iT - 19T^{2} \)
23 \( 1 + (-2.40 - 4.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.81 + 2.20i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.49 + 5.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.40 + 3.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.27 - 7.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.84 - 2.79i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0633 - 0.109i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.0 - 6.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.42T + 61T^{2} \)
67 \( 1 + 5.69iT - 67T^{2} \)
71 \( 1 + (7.24 - 4.18i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.21 - 3.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.798 + 1.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.39iT - 83T^{2} \)
89 \( 1 + (-11.6 + 6.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.75 - 5.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.88517806657664162467903456981, −9.818636242795746182602709758847, −9.101458428467349383097785997815, −8.672044195252082494433730003341, −7.41915624182826212346968981132, −6.51078724015612967106933404027, −5.79815056183013432556552520000, −4.41154484876142676062113871277, −2.76790417655164031647976071310, −1.87894483838216314728078227332, 0.951280497260900296398138336028, 2.41599660589959933451007700434, 3.58175325309237199822713962934, 4.79700671124861559595460700550, 6.15251592862173345903231523535, 7.25387137921262697674110010274, 8.036435117981272639908147725623, 8.918202765724891316456698998694, 9.895394858371194433517008821761, 10.20520739750842532707496945950

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