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Properties

Label 2-546-91.88-c1-0-6
Degree $2$
Conductor $546$
Sign $0.595 - 0.803i$
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  + (0.866 + 0.5i)2-s − 3-s + (0.499 + 0.866i)4-s + (−0.594 + 0.343i)5-s + (−0.866 − 0.5i)6-s + (2.63 + 0.246i)7-s + 0.999i·8-s + 9-s − 0.686·10-s − 4.38i·11-s + (−0.499 − 0.866i)12-s + (3.21 + 1.62i)13-s + (2.15 + 1.53i)14-s + (0.594 − 0.343i)15-s + (−0.5 + 0.866i)16-s + (2.03 + 3.52i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s − 0.577·3-s + (0.249 + 0.433i)4-s + (−0.265 + 0.153i)5-s + (−0.353 − 0.204i)6-s + (0.995 + 0.0931i)7-s + 0.353i·8-s + 0.333·9-s − 0.216·10-s − 1.32i·11-s + (−0.144 − 0.249i)12-s + (0.892 + 0.451i)13-s + (0.576 + 0.409i)14-s + (0.153 − 0.0885i)15-s + (−0.125 + 0.216i)16-s + (0.493 + 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65934 + 0.834959i\)
\(L(\frac12)\) \(\approx\) \(1.65934 + 0.834959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + T \)
7 \( 1 + (-2.63 - 0.246i)T \)
13 \( 1 + (-3.21 - 1.62i)T \)
good5 \( 1 + (0.594 - 0.343i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.38iT - 11T^{2} \)
17 \( 1 + (-2.03 - 3.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 7.17iT - 19T^{2} \)
23 \( 1 + (-0.862 + 1.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.181 + 0.313i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.49 - 2.01i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.49 + 3.17i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.74 + 3.31i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 + 4.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.38 + 5.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.12 - 1.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.16 - 1.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.04T + 61T^{2} \)
67 \( 1 + 3.84iT - 67T^{2} \)
71 \( 1 + (13.3 + 7.70i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.0 + 5.77i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.43 - 2.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.79iT - 83T^{2} \)
89 \( 1 + (7.10 + 4.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.62 - 1.51i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.94632242702130102508432965912, −10.54344314479057257866423929451, −8.855955103862106318384967415916, −8.164581132741413384311037841453, −7.29121412268854973366344751070, −5.89719195923005972001629907604, −5.72762306414481128957388731360, −4.26438901563687244989175405231, −3.47398165527398310947331702897, −1.54904658840062278290561864307, 1.15862143308814748377867151411, 2.66880367279473077278535098296, 4.31249104622386505819468449827, 4.80676580706843196790160862814, 5.82964285221883279901857750752, 7.02766424173627290273929810163, 7.77639780348531031935175371619, 9.030684352727147226000727723202, 10.05543189538915136610003444697, 10.92007135334744500209337779882

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