Properties

Label 2-546-273.194-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.946 - 0.322i$
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 + (1.24 + 1.20i)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 + (0.105 + 2.99i)9-s + (−0.0916 + 0.0529i)10-s + (0.603 + 1.04i)11-s + (0.418 − 1.68i)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 + (0.719 + 0.694i)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 + (0.0353 + 0.999i)9-s + (−0.0289 + 0.0167i)10-s + (0.182 + 0.315i)11-s + (0.120 − 0.485i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.946 - 0.322i$
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: \((2,\ 546,\ (\ :1/2),\ -0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175056 + 1.05813i\)
\(L(\frac12)\) \(\approx\) \(0.175056 + 1.05813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 - 1.20i)T \)
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

−10.77541431095373230560768970030, −9.972620791685662462513645488354, −9.465135758649933656709268845191, −8.607559853043472119408203665498, −7.77059120082135008721798988891, −6.75768374650441164502374567587, −5.70506569229844442387089905381, −4.65243521732860809227050919615, −3.49166612243848754399199228057, −2.20403102979667474181754653756, 0.61978000348222096193365407396, 2.31176514919982493322469723389, 3.22111246071999708449834009120, 4.33311734993307302969491986559, 5.98702532710896065641720017870, 7.11853869993067684718601301330, 7.65807261683875885365463742493, 8.835832248744330662685596125521, 9.513012644642818913566040333181, 10.17632855796760017204015107307

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