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Properties

Label 2-546-273.200-c1-0-19
Degree $2$
Conductor $546$
Sign $0.988 + 0.152i$
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  + (0.258 − 0.965i)2-s + (0.562 + 1.63i)3-s + (−0.866 − 0.499i)4-s + (0.312 − 1.16i)5-s + (1.72 − 0.119i)6-s + (2.64 + 0.138i)7-s + (−0.707 + 0.707i)8-s + (−2.36 + 1.84i)9-s + (−1.04 − 0.604i)10-s + (0.747 + 2.78i)11-s + (0.331 − 1.70i)12-s + (2.87 − 2.17i)13-s + (0.817 − 2.51i)14-s + (2.08 − 0.144i)15-s + (0.500 + 0.866i)16-s + (0.862 − 1.49i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.324 + 0.945i)3-s + (−0.433 − 0.249i)4-s + (0.139 − 0.522i)5-s + (0.705 − 0.0488i)6-s + (0.998 + 0.0522i)7-s + (−0.249 + 0.249i)8-s + (−0.788 + 0.614i)9-s + (−0.330 − 0.191i)10-s + (0.225 + 0.840i)11-s + (0.0957 − 0.490i)12-s + (0.798 − 0.602i)13-s + (0.218 − 0.672i)14-s + (0.539 − 0.0373i)15-s + (0.125 + 0.216i)16-s + (0.209 − 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92163 - 0.147555i\)
\(L(\frac12)\) \(\approx\) \(1.92163 - 0.147555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.562 - 1.63i)T \)
7 \( 1 + (-2.64 - 0.138i)T \)
13 \( 1 + (-2.87 + 2.17i)T \)
good5 \( 1 + (-0.312 + 1.16i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.747 - 2.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.862 + 1.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.61 - 0.433i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.390 + 0.675i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.00iT - 29T^{2} \)
31 \( 1 + (-2.56 - 9.58i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.48 + 5.54i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.58 + 6.58i)T + 41iT^{2} \)
43 \( 1 - 1.13iT - 43T^{2} \)
47 \( 1 + (-5.05 - 1.35i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.11 + 3.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.89 - 10.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.35 - 5.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.36 + 8.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.22 + 1.22i)T + 71iT^{2} \)
73 \( 1 + (13.7 - 3.67i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (17.6 + 4.73i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.27 - 5.27i)T - 97iT^{2} \)
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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

−10.64674316515558479756692636408, −10.10699793522619539397511914060, −8.984248409548429564372951208142, −8.567640894149483154842019600738, −7.41210282362074547617155431078, −5.65541314632707496939343926119, −4.93362874641887635504812829654, −4.13424168657156836955229790676, −2.93560664962063608688296752228, −1.49427673037067233265778362714, 1.34410097789849279901419754019, 2.90136006825126122174846533314, 4.14432623019729690409533778241, 5.57744094322894098687659745848, 6.37115484665024445969989672243, 7.14287187534706903057610123941, 8.231113268516215477864838808675, 8.518890511163059509003195231226, 9.757863381412503033343151875167, 11.19426665969845470995976505628

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