Properties

Label 2-546-273.242-c1-0-22
Degree $2$
Conductor $546$
Sign $0.918 + 0.394i$
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  + (−0.965 − 0.258i)2-s + (1.19 − 1.24i)3-s + (0.866 + 0.499i)4-s + (4.04 + 1.08i)5-s + (−1.48 + 0.896i)6-s + (−1.05 + 2.42i)7-s + (−0.707 − 0.707i)8-s + (−0.124 − 2.99i)9-s + (−3.62 − 2.09i)10-s + (5.02 − 1.34i)11-s + (1.66 − 0.482i)12-s + (−2.74 − 2.33i)13-s + (1.65 − 2.06i)14-s + (6.20 − 3.75i)15-s + (0.500 + 0.866i)16-s + (−3.25 + 5.63i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.692 − 0.721i)3-s + (0.433 + 0.249i)4-s + (1.80 + 0.484i)5-s + (−0.604 + 0.366i)6-s + (−0.400 + 0.916i)7-s + (−0.249 − 0.249i)8-s + (−0.0413 − 0.999i)9-s + (−1.14 − 0.662i)10-s + (1.51 − 0.405i)11-s + (0.480 − 0.139i)12-s + (−0.762 − 0.647i)13-s + (0.441 − 0.552i)14-s + (1.60 − 0.969i)15-s + (0.125 + 0.216i)16-s + (−0.788 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.918 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72512 - 0.354603i\)
\(L(\frac12)\) \(\approx\) \(1.72512 - 0.354603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.19 + 1.24i)T \)
7 \( 1 + (1.05 - 2.42i)T \)
13 \( 1 + (2.74 + 2.33i)T \)
good5 \( 1 + (-4.04 - 1.08i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-5.02 + 1.34i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.25 - 5.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0105 + 0.0393i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.836 - 1.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.23iT - 29T^{2} \)
31 \( 1 + (-1.08 + 0.291i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.25 + 1.94i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.50 - 2.50i)T - 41iT^{2} \)
43 \( 1 + 2.91iT - 43T^{2} \)
47 \( 1 + (-1.12 + 4.20i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.06 + 1.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.99 + 1.87i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.51 - 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.09 + 1.36i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.64 - 1.64i)T - 71iT^{2} \)
73 \( 1 + (3.62 + 13.5i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.80 + 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.53 - 9.53i)T - 83iT^{2} \)
89 \( 1 + (-2.05 + 7.67i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.41 + 6.41i)T + 97iT^{2} \)
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

−10.42121874017164773665870876443, −9.667350887077309862879950739827, −9.021779502594733742571692413806, −8.452797132863623848739377373112, −6.94513837641112702429079463352, −6.39132016458712982532457851098, −5.65572776997098363236386910042, −3.43214900863463093144122071797, −2.36376584830137606469833651495, −1.61336597085681799614116284323, 1.52216604796827128725119617682, 2.65003190875354258568506492230, 4.29512066248118945255519621967, 5.19143798167520044636712108275, 6.60723046213755810224036336521, 7.08465410271337041665551627659, 8.685904460051561830105005098735, 9.310956184508704415862084330302, 9.708439015834910718848517388385, 10.33382738782128743159459168774

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