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Properties

Label 2-546-273.44-c1-0-7
Degree $2$
Conductor $546$
Sign $0.679 - 0.734i$
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  + (−0.965 + 0.258i)2-s + (0.910 − 1.47i)3-s + (0.866 − 0.499i)4-s + (−2.92 + 0.782i)5-s + (−0.497 + 1.65i)6-s + (−0.584 + 2.58i)7-s + (−0.707 + 0.707i)8-s + (−1.34 − 2.68i)9-s + (2.61 − 1.51i)10-s + (1.60 + 0.429i)11-s + (0.0513 − 1.73i)12-s + (3.59 + 0.238i)13-s + (−0.103 − 2.64i)14-s + (−1.50 + 5.01i)15-s + (0.500 − 0.866i)16-s + (0.580 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.525 − 0.850i)3-s + (0.433 − 0.249i)4-s + (−1.30 + 0.350i)5-s + (−0.203 + 0.677i)6-s + (−0.220 + 0.975i)7-s + (−0.249 + 0.249i)8-s + (−0.447 − 0.894i)9-s + (0.828 − 0.478i)10-s + (0.483 + 0.129i)11-s + (0.0148 − 0.499i)12-s + (0.997 + 0.0660i)13-s + (−0.0275 − 0.706i)14-s + (−0.388 + 1.29i)15-s + (0.125 − 0.216i)16-s + (0.140 + 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.679 - 0.734i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.679 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.830285 + 0.362948i\)
\(L(\frac12)\) \(\approx\) \(0.830285 + 0.362948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.910 + 1.47i)T \)
7 \( 1 + (0.584 - 2.58i)T \)
13 \( 1 + (-3.59 - 0.238i)T \)
good5 \( 1 + (2.92 - 0.782i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.60 - 0.429i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.580 - 1.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.29 - 4.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.67 - 2.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.25iT - 29T^{2} \)
31 \( 1 + (-9.23 - 2.47i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.84 + 1.29i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.39 - 1.39i)T + 41iT^{2} \)
43 \( 1 + 0.835iT - 43T^{2} \)
47 \( 1 + (-0.406 - 1.51i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.00 + 5.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 + 0.575i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.85 - 4.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.8 + 3.44i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.08 - 4.08i)T + 71iT^{2} \)
73 \( 1 + (3.80 - 14.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.33 + 5.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.280 + 0.280i)T + 83iT^{2} \)
89 \( 1 + (2.01 + 7.51i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.91 - 7.91i)T - 97iT^{2} \)
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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

−11.06220226566769374685121513526, −9.834519298851607274013495674329, −8.792226681575320303250740741029, −8.277374050921960750025119863665, −7.53244257211782290073284546621, −6.59769747974427516109384345728, −5.79662189427501516669878345087, −3.86566932778001431483851922833, −2.93847945247199299995871575499, −1.38672857893724524226278742244, 0.69493639384188245052861033288, 2.91010538616894150104599862952, 3.97234515380384042337787872267, 4.50611914119100800222968247972, 6.29784706920545606259775247906, 7.51870917322921382441842548412, 8.132383662440284641939391096063, 8.898795050565595455461571433452, 9.770392090342952399078044219993, 10.64636955437461475023214144799

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