Properties

Label 2-546-13.10-c1-0-4
Degree $2$
Conductor $546$
Sign $0.575 - 0.817i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + 3.05i·5-s + (0.866 + 0.499i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.52 + 2.64i)10-s + (−2.98 + 1.72i)11-s + 0.999·12-s + (3.25 − 1.55i)13-s + 0.999·14-s + (−2.64 + 1.52i)15-s + (−0.5 − 0.866i)16-s + (−1.41 + 2.44i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + 1.36i·5-s + (0.353 + 0.204i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.483 + 0.836i)10-s + (−0.898 + 0.518i)11-s + 0.288·12-s + (0.902 − 0.431i)13-s + 0.267·14-s + (−0.683 + 0.394i)15-s + (−0.125 − 0.216i)16-s + (−0.343 + 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95825 + 1.01573i\)
\(L(\frac12)\) \(\approx\) \(1.95825 + 1.01573i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.25 + 1.55i)T \)
good5 \( 1 - 3.05iT - 5T^{2} \)
11 \( 1 + (2.98 - 1.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.41 - 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.49 - 0.862i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.92 + 3.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.978iT - 31T^{2} \)
37 \( 1 + (-4.58 + 2.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.62 + 4.98i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.04iT - 47T^{2} \)
53 \( 1 + 8.33T + 53T^{2} \)
59 \( 1 + (-8.64 - 4.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.77 + 9.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.11 + 2.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.17 + 1.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 4.49T + 79T^{2} \)
83 \( 1 + 7.15iT - 83T^{2} \)
89 \( 1 + (6.84 - 3.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.88 + 4.54i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89294340653573189869796428081, −10.36055270479569596047627452523, −9.459399000979135741369637108894, −8.167924664204393500307798341725, −7.31572537034515420682385259354, −6.18486553817925258198178790297, −5.31198564109933182563469889970, −4.03818256803755228264191672854, −3.13249927276825195201384804925, −2.15183689530463916303899119836, 1.11541997439468983081592657501, 2.73310824370063157662183896347, 4.16483731699570470533197513691, 5.04892507829891282149478440926, 5.92160382064144849344900282171, 7.06573337410252472382046795962, 8.086954467108983644981362737650, 8.606839642397001830075488635440, 9.502790436806801175032102866839, 11.01241482286716819407308027184

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