↑

Properties

Label 2-546-21.5-c1-0-24
Degree $2$
Conductor $546$
Sign $0.914 + 0.403i$
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 + (0.691 − 1.58i)3-s + (0.499 + 0.866i)4-s + (0.735 − 1.27i)5-s + (1.39 − 1.02i)6-s + (0.322 + 2.62i)7-s + 0.999i·8-s + (−2.04 − 2.19i)9-s + (1.27 − 0.735i)10-s + (5.30 − 3.06i)11-s + (1.72 − 0.194i)12-s + i·13-s + (−1.03 + 2.43i)14-s + (−1.51 − 2.05i)15-s + (−0.5 + 0.866i)16-s + (−0.120 − 0.209i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.399 − 0.916i)3-s + (0.249 + 0.433i)4-s + (0.329 − 0.569i)5-s + (0.568 − 0.420i)6-s + (0.121 + 0.992i)7-s + 0.353i·8-s + (−0.680 − 0.732i)9-s + (0.403 − 0.232i)10-s + (1.59 − 0.923i)11-s + (0.496 − 0.0562i)12-s + 0.277i·13-s + (−0.276 + 0.650i)14-s + (−0.391 − 0.529i)15-s + (−0.125 + 0.216i)16-s + (−0.0293 − 0.0508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46589 - 0.519744i\)
\(L(\frac12)\) \(\approx\) \(2.46589 - 0.519744i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.691 + 1.58i)T \)
7 \( 1 + (-0.322 - 2.62i)T \)
13 \( 1 - iT \)
good5 \( 1 + (-0.735 + 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.30 + 3.06i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.120 + 0.209i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.75 + 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.69 - 0.976i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.00iT - 29T^{2} \)
31 \( 1 + (0.975 - 0.562i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.60 + 4.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.60T + 41T^{2} \)
43 \( 1 - 1.24T + 43T^{2} \)
47 \( 1 + (3.84 - 6.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.14 - 3.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.21 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.5 - 6.65i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.58 - 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.38iT - 71T^{2} \)
73 \( 1 + (8.77 - 5.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.38 + 4.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.45T + 83T^{2} \)
89 \( 1 + (1.82 - 3.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.44iT - 97T^{2} \)
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

−11.32397258270854931390179765716, −9.330669700596899894644826251727, −8.889155955370798610834492500937, −8.157744499171606729373287538687, −6.85783932164891112011302253686, −6.21492250016243752196910099555, −5.38514969242708468116932035095, −4.00783814439230781658128268599, −2.75682830349596898801067126065, −1.46608897122390562491961556118, 1.81957863189346387531653200133, 3.26409445657810495511855699837, 4.11797230703793133786784534928, 4.84417131032199822783981586126, 6.29724123157240541362592139752, 7.01527679650423985616810882152, 8.315195320648254708140976506854, 9.440699663860991797263896546097, 10.13689666063309363784412506474, 10.72395550125940185883711616701

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