↑

Properties

Label 2-546-13.4-c1-0-3
Degree $2$
Conductor $546$
Sign $0.265 - 0.964i$
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 + 2.73i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.36 + 2.36i)10-s + (0.232 + 0.133i)11-s + 0.999·12-s + (0.866 + 3.5i)13-s − 0.999·14-s + (2.36 + 1.36i)15-s + (−0.5 + 0.866i)16-s + (1.86 + 3.23i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 1.22i·5-s + (0.353 − 0.204i)6-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.431 + 0.748i)10-s + (0.0699 + 0.0403i)11-s + 0.288·12-s + (0.240 + 0.970i)13-s − 0.267·14-s + (0.610 + 0.352i)15-s + (−0.125 + 0.216i)16-s + (0.452 + 0.783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69309 + 1.29055i\)
\(L(\frac12)\) \(\approx\) \(1.69309 + 1.29055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 3.5i)T \)
good5 \( 1 - 2.73iT - 5T^{2} \)
11 \( 1 + (-0.232 - 0.133i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.86 - 3.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.13 + 1.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.66iT - 31T^{2} \)
37 \( 1 + (4.09 + 2.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.06 - 3.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.36 + 2.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.46iT - 47T^{2} \)
53 \( 1 + 3.53T + 53T^{2} \)
59 \( 1 + (-11.1 + 6.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.13 - 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.803 + 0.464i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 4.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (-0.401 - 0.232i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 1.36i)T + (48.5 - 84.0i)T^{2} \)
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

−11.26219129420630923322483571580, −10.12573466100023491219879371664, −9.188286337130182951260893280732, −7.998324770354911891776125662998, −7.22502593975215821859943862949, −6.44132324408746688157840271153, −5.76025197894167119172042105702, −4.12979556107327993831553293039, −3.20087625567346718968660012328, −2.09372652888537309045567056225, 1.06637311802307695459857089553, 2.87647155089616607705287167428, 3.87355471689810194775368080787, 4.99532461089935335406458173485, 5.51732571543058982983138659094, 6.91614274089525594764423683270, 8.172679866592980071120532380470, 8.907991890266408844720914303536, 9.882957221886215943366298830388, 10.50427805019002074871862228498

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