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Properties

Label 2-546-7.2-c1-0-15
Degree $2$
Conductor $546$
Sign $-0.991 - 0.126i$
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 + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1 − 1.73i)5-s − 0.999·6-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (−1.5 − 2.59i)11-s + (−0.499 + 0.866i)12-s − 13-s + (−0.500 + 2.59i)14-s − 1.99·15-s + (−0.5 + 0.866i)16-s + (−2.5 − 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s − 0.408·6-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (−0.452 − 0.783i)11-s + (−0.144 + 0.249i)12-s − 0.277·13-s + (−0.133 + 0.694i)14-s − 0.516·15-s + (−0.125 + 0.216i)16-s + (−0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0620395 + 0.977614i\)
\(L(\frac12)\) \(\approx\) \(0.0620395 + 0.977614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
13 \( 1 + T \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−10.44207720444818136592454567071, −9.406343746102047802525323498436, −8.908850760181139815147167552718, −7.62525600952476278023484431714, −6.40540042738356625626577830248, −5.63331278480893947610050772058, −4.77815757539992127924390644440, −3.28162877485795102032516129313, −2.16562012565718001178786997235, −0.49527662980868924351165593227, 2.52880618990723688704929060118, 3.71487970712276797995271084164, 4.73291315000348340617855276648, 5.90676224648613412199363688981, 6.62312608806841709158313106572, 7.34544332234098272794883785145, 8.622139728649890942531257274479, 9.686186313212732169722741890781, 10.28313845173251812869786769390, 11.03779459159792050806087919746

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