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Properties

Label 2-546-273.38-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.768 - 0.640i$
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 + (1.40 + 1.00i)3-s + (−0.499 + 0.866i)4-s + (−2.19 + 1.26i)5-s + (−0.170 + 1.72i)6-s + (2.61 + 0.373i)7-s − 0.999·8-s + (0.962 + 2.84i)9-s + (−2.19 − 1.26i)10-s + (−1.32 + 2.29i)11-s + (−1.57 + 0.714i)12-s + (−3.59 − 0.207i)13-s + (0.986 + 2.45i)14-s + (−4.37 − 0.432i)15-s + (−0.5 − 0.866i)16-s + (1.84 − 3.19i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.812 + 0.582i)3-s + (−0.249 + 0.433i)4-s + (−0.982 + 0.567i)5-s + (−0.0695 + 0.703i)6-s + (0.989 + 0.141i)7-s − 0.353·8-s + (0.320 + 0.947i)9-s + (−0.694 − 0.401i)10-s + (−0.399 + 0.691i)11-s + (−0.455 + 0.206i)12-s + (−0.998 − 0.0575i)13-s + (0.263 + 0.656i)14-s + (−1.12 − 0.111i)15-s + (−0.125 − 0.216i)16-s + (0.447 − 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.768 - 0.640i$
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: \((2,\ 546,\ (\ :1/2),\ -0.768 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635986 + 1.75703i\)
\(L(\frac12)\) \(\approx\) \(0.635986 + 1.75703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 1.00i)T \)
7 \( 1 + (-2.61 - 0.373i)T \)
13 \( 1 + (3.59 + 0.207i)T \)
good5 \( 1 + (2.19 - 1.26i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.32 - 2.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.84 + 3.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.241 - 0.418i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.09 + 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.93iT - 29T^{2} \)
31 \( 1 + (-2.71 + 4.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.90 - 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.03iT - 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 + (-6.66 + 3.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.13 + 4.69i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.98 - 2.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.25 + 2.45i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 0.797i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-6.57 + 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.75 + 6.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.10iT - 83T^{2} \)
89 \( 1 + (-2.37 + 1.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−11.13299716873564714634471276211, −10.22587084550398463012578196917, −9.224842686312661821132062076140, −8.237102593603927365659883531824, −7.55222467221175672863779965486, −7.05277540760532259425284446305, −5.14064109165513668678711652516, −4.69064929845692256979628420481, −3.51158265668511585766153334315, −2.45167578673392755440367993752, 0.932585516414835373424842767549, 2.35802252450530514608160326788, 3.60480700122091254764198351934, 4.50130960305693747868408775319, 5.57513917833339733343218116605, 7.12602620674392414933495853346, 7.987406904687830512968067815778, 8.485277044335388873614683494334, 9.500015165337589032324338366199, 10.65181488148534434478034727145

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