Properties

Label 2-546-273.38-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.669 - 0.742i$
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 + (−1.63 − 0.568i)3-s + (−0.499 + 0.866i)4-s + (1.08 − 0.627i)5-s + (−0.326 − 1.70i)6-s + (−1.14 + 2.38i)7-s − 0.999·8-s + (2.35 + 1.85i)9-s + (1.08 + 0.627i)10-s + (−0.900 + 1.56i)11-s + (1.31 − 1.13i)12-s + (0.657 − 3.54i)13-s + (−2.63 + 0.204i)14-s + (−2.13 + 0.409i)15-s + (−0.5 − 0.866i)16-s + (−1.12 + 1.95i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.944 − 0.328i)3-s + (−0.249 + 0.433i)4-s + (0.486 − 0.280i)5-s + (−0.133 − 0.694i)6-s + (−0.431 + 0.902i)7-s − 0.353·8-s + (0.784 + 0.619i)9-s + (0.343 + 0.198i)10-s + (−0.271 + 0.470i)11-s + (0.378 − 0.327i)12-s + (0.182 − 0.983i)13-s + (−0.704 + 0.0545i)14-s + (−0.551 + 0.105i)15-s + (−0.125 − 0.216i)16-s + (−0.273 + 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.385173 + 0.865735i\)
\(L(\frac12)\) \(\approx\) \(0.385173 + 0.865735i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 0.568i)T \)
7 \( 1 + (1.14 - 2.38i)T \)
13 \( 1 + (-0.657 + 3.54i)T \)
good5 \( 1 + (-1.08 + 0.627i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.900 - 1.56i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.12 - 1.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.46 - 4.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.93 - 4.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.62iT - 29T^{2} \)
31 \( 1 + (2.39 - 4.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.77 + 2.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.52iT - 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 + (-8.20 + 4.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.2 + 6.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.80 + 2.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.64 + 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 - 6.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.21T + 71T^{2} \)
73 \( 1 + (-7.95 + 13.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.323 + 0.560i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + (-11.1 + 6.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.48T + 97T^{2} \)
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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

−11.27243051988215364438612834725, −10.14846424386326657912140947241, −9.467754937480554836463315306473, −8.200136420792010027455693706107, −7.45030713679237486221141773579, −6.21401403764440726538671454392, −5.71036070069480253817866786550, −5.02686632781647135849690845217, −3.52915115846446261079192046012, −1.81794016094303361650712138621, 0.54185454699465663027934007005, 2.37722427235234403532987430754, 3.93275334450370313986150258443, 4.56781437537459912458102354690, 5.93750028310530502136538559155, 6.44767982713684841229025415915, 7.58578098066498699952124578569, 9.208870826381877367705017924824, 9.855566910992162504836581647890, 10.59799312351022108385768699404

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