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Properties

Label 2-546-273.194-c1-0-18
Degree $2$
Conductor $546$
Sign $0.553 + 0.832i$
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  + (−0.5 + 0.866i)2-s + (−0.418 − 1.68i)3-s + (−0.499 − 0.866i)4-s + (0.0916 + 0.0529i)5-s + (1.66 + 0.477i)6-s + (2.29 − 1.31i)7-s + 0.999·8-s + (−2.64 + 1.40i)9-s + (−0.0916 + 0.0529i)10-s + (0.603 + 1.04i)11-s + (−1.24 + 1.20i)12-s + (3.60 + 0.175i)13-s + (−0.0147 + 2.64i)14-s + (0.0505 − 0.176i)15-s + (−0.5 + 0.866i)16-s + (−1.14 − 1.97i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.241 − 0.970i)3-s + (−0.249 − 0.433i)4-s + (0.0409 + 0.0236i)5-s + (0.679 + 0.195i)6-s + (0.868 − 0.495i)7-s + 0.353·8-s + (−0.883 + 0.469i)9-s + (−0.0289 + 0.0167i)10-s + (0.182 + 0.315i)11-s + (−0.359 + 0.347i)12-s + (0.998 + 0.0485i)13-s + (−0.00394 + 0.707i)14-s + (0.0130 − 0.0454i)15-s + (−0.125 + 0.216i)16-s + (−0.276 − 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00570 - 0.538981i\)
\(L(\frac12)\) \(\approx\) \(1.00570 - 0.538981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.418 + 1.68i)T \)
7 \( 1 + (-2.29 + 1.31i)T \)
13 \( 1 + (-3.60 - 0.175i)T \)
good5 \( 1 + (-0.0916 - 0.0529i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.603 - 1.04i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.14 + 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.82 + 3.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.845 + 0.488i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.03iT - 29T^{2} \)
31 \( 1 + (0.610 + 1.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.01 + 1.16i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.417iT - 41T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 + (-1.47 - 0.854i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.77 + 1.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.04 + 5.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.67 + 5.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.65 - 3.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 + (-3.72 - 6.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.04 - 8.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + (-14.7 - 8.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.85T + 97T^{2} \)
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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

−10.84151578114245468016814192969, −9.646079486823519183226845596067, −8.565660911879906029596688129155, −7.908894910493487037670531878250, −7.10253992541301235797768857412, −6.29303781842037317731683095053, −5.31242679165745013766747258551, −4.16633475747743754263166239225, −2.22186539338390375498215293589, −0.854525430418941867309303612480, 1.55111022333169848138394275098, 3.20746651429519854711560938232, 4.10813925617175351882421347327, 5.26185361208732970256273159452, 6.07287622770327730934642702291, 7.67316771562311632020322115931, 8.729043016497274654490385287340, 9.057174127570664317226073147522, 10.28131212416563433023015932124, 10.86479739103339641274252825385

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