↑

Properties

Label 2-546-7.2-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.198 - 0.980i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 0.999·6-s + (−1.62 − 2.09i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (1.20 + 2.09i)11-s + (−0.499 + 0.866i)12-s + 13-s + (2.62 − 0.358i)14-s + 1.41·15-s + (−0.5 + 0.866i)16-s + (1.62 + 2.80i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.316 + 0.547i)5-s + 0.408·6-s + (−0.612 − 0.790i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.223 − 0.387i)10-s + (0.363 + 0.630i)11-s + (−0.144 + 0.249i)12-s + 0.277·13-s + (0.700 − 0.0958i)14-s + 0.365·15-s + (−0.125 + 0.216i)16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467267 + 0.571127i\)
\(L(\frac12)\) \(\approx\) \(0.467267 + 0.571127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.62 + 2.09i)T \)
13 \( 1 - T \)
good5 \( 1 + (0.707 - 1.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.20 - 2.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.62 - 2.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.53 - 4.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (0.585 + 1.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.53 - 7.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
43 \( 1 + 0.828T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.15 - 8.93i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.20 + 9.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.62 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.67 - 2.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (6.53 + 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.65 + 8.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.65T + 83T^{2} \)
89 \( 1 + (1.12 - 1.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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

−10.83056680561678158347385980964, −10.20493216616256174298252191608, −9.308846801732036712497495372024, −8.059016096855535551211125899199, −7.41807386623546763877067190133, −6.60042115296952988465668071790, −5.93238323097838767289526698454, −4.46984593446111680159254729091, −3.34003485197684301604645153736, −1.42137775860113999582923988484, 0.53866695850370366748052411926, 2.55046676697355003831634042006, 3.68391471261339286920967007639, 4.75495220560467322618676249032, 5.82688353516196206132468835604, 6.87877455309099440099068956198, 8.408406625103473051569127073908, 8.808198181993392778920449009412, 9.687180881092658075457870604125, 10.54383305658221336821938592985

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