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Properties

Label 2-546-273.185-c1-0-25
Degree $2$
Conductor $546$
Sign $0.162 + 0.986i$
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 + (−0.266 + 1.71i)3-s + (0.499 + 0.866i)4-s + (0.279 + 0.483i)5-s + (1.08 − 1.34i)6-s + (0.253 − 2.63i)7-s − 0.999i·8-s + (−2.85 − 0.912i)9-s − 0.558i·10-s − 6.43i·11-s + (−1.61 + 0.624i)12-s + (−1.74 + 3.15i)13-s + (−1.53 + 2.15i)14-s + (−0.902 + 0.349i)15-s + (−0.5 + 0.866i)16-s + (−3.19 − 5.53i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.153 + 0.988i)3-s + (0.249 + 0.433i)4-s + (0.124 + 0.216i)5-s + (0.443 − 0.550i)6-s + (0.0957 − 0.995i)7-s − 0.353i·8-s + (−0.952 − 0.304i)9-s − 0.176i·10-s − 1.94i·11-s + (−0.466 + 0.180i)12-s + (−0.483 + 0.875i)13-s + (−0.410 + 0.575i)14-s + (−0.233 + 0.0901i)15-s + (−0.125 + 0.216i)16-s + (−0.775 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.162 + 0.986i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.162 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554851 - 0.470732i\)
\(L(\frac12)\) \(\approx\) \(0.554851 - 0.470732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.266 - 1.71i)T \)
7 \( 1 + (-0.253 + 2.63i)T \)
13 \( 1 + (1.74 - 3.15i)T \)
good5 \( 1 + (-0.279 - 0.483i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 6.43iT - 11T^{2} \)
17 \( 1 + (3.19 + 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.04iT - 19T^{2} \)
23 \( 1 + (-0.412 - 0.237i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.00 - 1.73i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.995 - 0.574i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.01 + 8.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.45 - 5.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.50 + 2.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.145 + 0.251i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.18 + 4.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.85 - 8.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.08iT - 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + (7.85 + 4.53i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.17 + 5.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.89T + 83T^{2} \)
89 \( 1 + (-1.61 + 2.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.63 + 2.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.77836130751506859779323591118, −9.697294718194273969654560181354, −9.087666012965531643339405671809, −8.193702415677570708590411392216, −7.04062218864251275385249877585, −6.05321367648680540677289797524, −4.74014438209831943754383770706, −3.74508634677603085845085252108, −2.70658413041493645798263922793, −0.51692479194640489002771474033, 1.65366509592219070183612427518, 2.55876897444086681688325356284, 4.73467875747733681940744035032, 5.68593781727197325500093098040, 6.59603711735572354641844242333, 7.47851778834014057250375346273, 8.217942065424195057946076743589, 9.109032007085829199112559388731, 9.948345718374711162144672551524, 11.00222089047939574039570818935

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