Properties

Label 2-546-273.62-c1-0-29
Degree $2$
Conductor $546$
Sign $-0.977 - 0.213i$
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 + (−1.18 − 1.26i)3-s + (−0.499 + 0.866i)4-s − 0.792i·5-s + (−0.5 + 1.65i)6-s + (2.5 − 0.866i)7-s + 0.999·8-s + (−0.186 + 2.99i)9-s + (−0.686 + 0.396i)10-s + (−2.18 − 3.78i)11-s + (1.68 − 0.396i)12-s + (−3.5 − 0.866i)13-s + (−2 − 1.73i)14-s + (−1 + 0.939i)15-s + (−0.5 − 0.866i)16-s + (2.18 − 3.78i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.684 − 0.728i)3-s + (−0.249 + 0.433i)4-s − 0.354i·5-s + (−0.204 + 0.677i)6-s + (0.944 − 0.327i)7-s + 0.353·8-s + (−0.0620 + 0.998i)9-s + (−0.216 + 0.125i)10-s + (−0.659 − 1.14i)11-s + (0.486 − 0.114i)12-s + (−0.970 − 0.240i)13-s + (−0.534 − 0.462i)14-s + (−0.258 + 0.242i)15-s + (−0.125 − 0.216i)16-s + (0.530 − 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.977 - 0.213i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.977 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0682331 + 0.633000i\)
\(L(\frac12)\) \(\approx\) \(0.0682331 + 0.633000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 1.26i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
13 \( 1 + (3.5 + 0.866i)T \)
good5 \( 1 + 0.792iT - 5T^{2} \)
11 \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.18 + 2.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.68 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.18 - 1.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 + (10.1 - 5.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.18 - 4.72i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.939iT - 47T^{2} \)
53 \( 1 - 2.22iT - 53T^{2} \)
59 \( 1 + (5.31 + 3.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 5.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.05 + 13.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 9.62T + 79T^{2} \)
83 \( 1 - 1.58iT - 83T^{2} \)
89 \( 1 + (9.30 - 5.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.372 + 0.644i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.67624715171640553261291751918, −9.565294721918741011296529708882, −8.381517559335769011946216015863, −7.77209431337103679357293691406, −6.92655921548222456488448209823, −5.32258941404889102390313224497, −4.98211054160208841873850350271, −3.18597415995688318318606313287, −1.79035854669682259963359720508, −0.44578731320238724266552267906, 1.99450657905141344499325334863, 3.92063457148674118812806012423, 5.03586786604847722971286312233, 5.53646853375080175619108399257, 6.82551194206921806856566443833, 7.61141488750175650607224273534, 8.617949492185138340879946278381, 9.657142016348176602972845026222, 10.33955603857537753581796758095, 10.95377306254011928489423664216

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