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Properties

Label 2-546-273.38-c1-0-21
Degree $2$
Conductor $546$
Sign $0.911 - 0.412i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.911 - 0.412i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.911 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26861 + 0.489493i\)
\(L(\frac12)\) \(\approx\) \(2.26861 + 0.489493i\)
\(L(\frac{3}{2})\) 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

−11.07695347550892118104987128959, −9.653692071512569267267612928072, −8.833841229917253538465049570862, −7.959597802632148890016232558355, −7.50183792836565257283487888734, −6.29181259744655496052837900666, −5.46516834609190756400692169840, −4.17206675111696633559575475587, −3.01068423046203609913671264309, −1.60209038129266371237577510648, 1.52490281160205422097273972304, 2.97310560127781345478634884191, 3.95487912082340406873418650503, 4.78210563512221455798909549081, 5.81882920874218819114064134773, 7.34305676597999907573673418124, 8.341973513547312119040580559411, 8.972299993080608112992844199614, 10.04887266499251450915483556761, 10.85368376163927628042456358972

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