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Properties

Label 2-546-91.88-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.215 - 0.976i$
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 + (−3.28 + 1.89i)5-s + (0.866 + 0.5i)6-s + (2.41 + 1.07i)7-s + 0.999i·8-s + 9-s − 3.79·10-s − 0.558i·11-s + (0.499 + 0.866i)12-s + (−2.73 + 2.35i)13-s + (1.55 + 2.13i)14-s + (−3.28 + 1.89i)15-s + (−0.5 + 0.866i)16-s + (3.02 + 5.24i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + 0.577·3-s + (0.249 + 0.433i)4-s + (−1.46 + 0.848i)5-s + (0.353 + 0.204i)6-s + (0.914 + 0.405i)7-s + 0.353i·8-s + 0.333·9-s − 1.19·10-s − 0.168i·11-s + (0.144 + 0.249i)12-s + (−0.758 + 0.652i)13-s + (0.416 + 0.571i)14-s + (−0.848 + 0.489i)15-s + (−0.125 + 0.216i)16-s + (0.734 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.215 - 0.976i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.215 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23998 + 1.54304i\)
\(L(\frac12)\) \(\approx\) \(1.23998 + 1.54304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 1.07i)T \)
13 \( 1 + (2.73 - 2.35i)T \)
good5 \( 1 + (3.28 - 1.89i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.558iT - 11T^{2} \)
17 \( 1 + (-3.02 - 5.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 4.21iT - 19T^{2} \)
23 \( 1 + (2.83 - 4.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 - 3.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.96 + 3.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.01 - 4.62i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.99 - 2.30i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.47 + 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.18 + 2.99i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.70 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.29 + 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.57T + 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + (4.58 + 2.64i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.29 - 1.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.38 - 7.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.48iT - 83T^{2} \)
89 \( 1 + (4.72 + 2.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.64 - 2.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34779547634439063530630780008, −10.37810705446883051846207031984, −8.995918213247143418701247974718, −8.051182749741847738902208742119, −7.56473674119732172112099201858, −6.73266234996604424493692828689, −5.35583593261871280003712094563, −4.20551183818781494827517426937, −3.50823342013794247508103059225, −2.23881997792669324054382888145, 0.940483751728791141048385755117, 2.68884523718739833185657162401, 4.00361523035692906799226412940, 4.54092432387070384182667428592, 5.52821034764826388586390423057, 7.41103773162126809446795993041, 7.71011833679333777562664106409, 8.617406045365969777677232211455, 9.741508897966709621127311576497, 10.71820983990983778446988038887

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