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Properties

Label 2-546-273.101-c1-0-30
Degree $2$
Conductor $546$
Sign $-0.157 + 0.987i$
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 + (1.60 − 0.646i)3-s + (−0.499 − 0.866i)4-s + (−2.60 + 1.50i)5-s + (−0.243 + 1.71i)6-s + (−0.916 − 2.48i)7-s + 0.999·8-s + (2.16 − 2.07i)9-s − 3.00i·10-s − 4.03·11-s + (−1.36 − 1.06i)12-s + (−0.138 − 3.60i)13-s + (2.60 + 0.446i)14-s + (−3.21 + 4.09i)15-s + (−0.5 + 0.866i)16-s + (−1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.927 − 0.373i)3-s + (−0.249 − 0.433i)4-s + (−1.16 + 0.672i)5-s + (−0.0992 + 0.700i)6-s + (−0.346 − 0.938i)7-s + 0.353·8-s + (0.721 − 0.692i)9-s − 0.950i·10-s − 1.21·11-s + (−0.393 − 0.308i)12-s + (−0.0382 − 0.999i)13-s + (0.696 + 0.119i)14-s + (−0.828 + 1.05i)15-s + (−0.125 + 0.216i)16-s + (−0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.157 + 0.987i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.157 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442621 - 0.518864i\)
\(L(\frac12)\) \(\approx\) \(0.442621 - 0.518864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 + 0.646i)T \)
7 \( 1 + (0.916 + 2.48i)T \)
13 \( 1 + (0.138 + 3.60i)T \)
good5 \( 1 + (2.60 - 1.50i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.03T + 11T^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.13T + 19T^{2} \)
23 \( 1 + (-2.14 - 1.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.74 + 2.16i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.95 + 5.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.22 + 1.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.29 - 3.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.23 - 3.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.77 + 3.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.50 + 4.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.75 + 1.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.66iT - 61T^{2} \)
67 \( 1 + 8.14iT - 67T^{2} \)
71 \( 1 + (-0.436 + 0.756i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.78 + 11.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.50 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + (3.15 + 1.81i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.10 - 1.91i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.43361830123603329652937785185, −9.665592385351913173637294877278, −8.299551108775080504784327424129, −7.915445977276994719911557962683, −7.18915786183942546000779548430, −6.45876454483451390030756214552, −4.77524232073423187966723511302, −3.66212397524129910250659967538, −2.67344668980434208758673851294, −0.37955266339975604343230119946, 2.04063992580048511239539074789, 3.12473059739107629032765198287, 4.23208074152222887800100416420, 4.99121971768176254238093472435, 6.77154011984651917004231481415, 7.982917371820314052747231339957, 8.615441345977717365019220431386, 8.965215794730701367123805204656, 10.19243147086597637670502354287, 10.88136365318740102409041670675

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