Properties

Label 2-546-91.47-c1-0-4
Degree $2$
Conductor $546$
Sign $0.691 - 0.722i$
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 + (−0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (0.140 + 0.522i)5-s + (−0.707 + 0.707i)6-s + (−2.24 + 1.39i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.270 + 0.468i)10-s + (2.61 + 0.700i)11-s + (−0.5 + 0.866i)12-s + (2.89 + 2.14i)13-s + (−1.80 + 1.93i)14-s + (−0.382 − 0.382i)15-s + (0.500 − 0.866i)16-s + (3.80 + 6.59i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (0.0626 + 0.233i)5-s + (−0.288 + 0.288i)6-s + (−0.849 + 0.527i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.0855 + 0.148i)10-s + (0.788 + 0.211i)11-s + (−0.144 + 0.249i)12-s + (0.803 + 0.594i)13-s + (−0.483 + 0.515i)14-s + (−0.0988 − 0.0988i)15-s + (0.125 − 0.216i)16-s + (0.922 + 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65392 + 0.706025i\)
\(L(\frac12)\) \(\approx\) \(1.65392 + 0.706025i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.24 - 1.39i)T \)
13 \( 1 + (-2.89 - 2.14i)T \)
good5 \( 1 + (-0.140 - 0.522i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.61 - 0.700i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.80 - 6.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.565 - 2.11i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.290 + 0.167i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.08T + 29T^{2} \)
31 \( 1 + (3.16 + 0.848i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.44 - 9.14i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.89 + 3.89i)T - 41iT^{2} \)
43 \( 1 + 5.36iT - 43T^{2} \)
47 \( 1 + (5.52 - 1.47i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.37 + 7.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.85 + 6.94i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-11.2 - 6.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.36 + 5.09i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.98 + 1.98i)T + 71iT^{2} \)
73 \( 1 + (-0.884 + 3.30i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.95 + 5.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.74 - 4.74i)T - 83iT^{2} \)
89 \( 1 + (17.9 - 4.81i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.03 + 9.03i)T - 97iT^{2} \)
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

−11.03387286244267971256896614280, −10.14507237210271277891089518696, −9.404198883270458614968332201376, −8.317236968640091699739622495165, −6.84748416284403155235497550169, −6.19773493715628181268419265735, −5.48931946773610437441088978821, −4.03896953265698106482935163115, −3.41050476804323325971345561193, −1.70224013317932346216803827474, 0.990474414238419838675189240009, 2.99575465177015816218438915108, 3.96165487246243558260721751134, 5.25060426823523305855226548113, 5.99201449563146699599585070685, 6.98600846696534632951472321824, 7.60087163356404738223234103289, 9.031885134904877683116875787209, 9.803492177650975270898774868812, 11.05936368956573940140482262379

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