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Properties

Label 2-546-273.185-c1-0-23
Degree $2$
Conductor $546$
Sign $-0.618 + 0.785i$
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 + (−1.18 − 1.26i)3-s + (0.499 + 0.866i)4-s + (0.386 + 0.669i)5-s + (0.388 + 1.68i)6-s + (0.662 − 2.56i)7-s − 0.999i·8-s + (−0.211 + 2.99i)9-s − 0.772i·10-s − 2.70i·11-s + (0.507 − 1.65i)12-s + (3.14 + 1.76i)13-s + (−1.85 + 1.88i)14-s + (0.391 − 1.28i)15-s + (−0.5 + 0.866i)16-s + (1.32 + 2.29i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.681 − 0.731i)3-s + (0.249 + 0.433i)4-s + (0.172 + 0.299i)5-s + (0.158 + 0.689i)6-s + (0.250 − 0.968i)7-s − 0.353i·8-s + (−0.0705 + 0.997i)9-s − 0.244i·10-s − 0.814i·11-s + (0.146 − 0.478i)12-s + (0.871 + 0.490i)13-s + (−0.495 + 0.504i)14-s + (0.101 − 0.330i)15-s + (−0.125 + 0.216i)16-s + (0.321 + 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.618 + 0.785i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343368 - 0.707536i\)
\(L(\frac12)\) \(\approx\) \(0.343368 - 0.707536i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 1.26i)T \)
7 \( 1 + (-0.662 + 2.56i)T \)
13 \( 1 + (-3.14 - 1.76i)T \)
good5 \( 1 + (-0.386 - 0.669i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.70iT - 11T^{2} \)
17 \( 1 + (-1.32 - 2.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 4.67iT - 19T^{2} \)
23 \( 1 + (4.66 + 2.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.22 + 2.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.24 + 1.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.28 - 3.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.90 + 8.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.66 - 9.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.63 + 4.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.16 + 1.82i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.61 + 9.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + 6.94T + 67T^{2} \)
71 \( 1 + (-9.71 - 5.60i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.99 - 5.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.14 - 8.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + (-5.17 + 8.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.86 - 3.96i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.73480095476690116778041239679, −9.861723403887686995732086551311, −8.457519589739708710770906035218, −7.971260575439809053168488502746, −6.68868813440867453300572368496, −6.35314769073425954390019199092, −4.83416107437515437492839996599, −3.51274895000820826970148877233, −1.95774712438491208078682881149, −0.64100576682271303833172301347, 1.56383401952106426784831966106, 3.40641263808988538624055261025, 4.89083041938624166646843925560, 5.58662591250517251200772461392, 6.38022250243749766959031948225, 7.64627331077400636896168147728, 8.667061747770421631029215841844, 9.358056080419400880941333085036, 10.15380328948857540062048766659, 10.89098095684958786588052612531

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