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Properties

Label 2-546-91.33-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.866 - 0.499i$
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 − i·3-s + (−0.866 − 0.499i)4-s + (−0.927 − 3.46i)5-s + (−0.965 − 0.258i)6-s + (−2.50 + 0.851i)7-s + (−0.707 + 0.707i)8-s − 9-s − 3.58·10-s + (2.81 − 2.81i)11-s + (−0.499 + 0.866i)12-s + (3.05 − 1.91i)13-s + (0.174 + 2.63i)14-s + (−3.46 + 0.927i)15-s + (0.500 + 0.866i)16-s + (−2.15 + 3.73i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s − 0.577i·3-s + (−0.433 − 0.249i)4-s + (−0.414 − 1.54i)5-s + (−0.394 − 0.105i)6-s + (−0.946 + 0.321i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s − 1.13·10-s + (0.849 − 0.849i)11-s + (−0.144 + 0.249i)12-s + (0.847 − 0.530i)13-s + (0.0466 + 0.705i)14-s + (−0.893 + 0.239i)15-s + (0.125 + 0.216i)16-s + (−0.523 + 0.906i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238220 + 0.890614i\)
\(L(\frac12)\) \(\approx\) \(0.238220 + 0.890614i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (2.50 - 0.851i)T \)
13 \( 1 + (-3.05 + 1.91i)T \)
good5 \( 1 + (0.927 + 3.46i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.81 + 2.81i)T - 11iT^{2} \)
17 \( 1 + (2.15 - 3.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.11 - 5.11i)T - 19iT^{2} \)
23 \( 1 + (0.616 - 0.356i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.49 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.21 + 1.93i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.856 + 0.229i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.511 - 1.91i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.1 + 5.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.60 - 0.430i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.862 + 1.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.91 - 1.04i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 8.13iT - 61T^{2} \)
67 \( 1 + (8.99 + 8.99i)T + 67iT^{2} \)
71 \( 1 + (0.167 - 0.623i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.01 - 3.77i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.49 - 4.32i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.15 + 7.15i)T - 83iT^{2} \)
89 \( 1 + (-4.33 + 16.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (8.33 + 2.23i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.39422767320966956112174618139, −9.175417262200379916608673874753, −8.664580515894435795638190789784, −7.982665757538917277621381138764, −6.17735061971839853691776306759, −5.84939679805167468471289513569, −4.25221601280507026954132097084, −3.55032564985235073582412101382, −1.81615349657984316603513912373, −0.50391710590918516168776721333, 2.75934597673346866368773323361, 3.76590375139647834851694401904, 4.54620652125858431317431210622, 6.18203569619764855719557783530, 6.85170064207837137626637485736, 7.24330309474394133575738112140, 8.861260505192618493845815214432, 9.396400688941670124068189489092, 10.57679266373824104426917793537, 11.02498432988053990431743489597

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