↑

Properties

Label 2-546-273.185-c1-0-34
Degree $2$
Conductor $546$
Sign $-0.891 - 0.452i$
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 + (−1.38 + 1.03i)3-s + (0.499 + 0.866i)4-s + (−1.75 − 3.03i)5-s + (1.72 − 0.201i)6-s + (1.37 − 2.26i)7-s − 0.999i·8-s + (0.858 − 2.87i)9-s + 3.50i·10-s + 3.53i·11-s + (−1.59 − 0.685i)12-s + (−3.59 + 0.316i)13-s + (−2.32 + 1.27i)14-s + (5.57 + 2.40i)15-s + (−0.5 + 0.866i)16-s + (1.97 + 3.42i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.801 + 0.597i)3-s + (0.249 + 0.433i)4-s + (−0.784 − 1.35i)5-s + (0.702 − 0.0823i)6-s + (0.519 − 0.854i)7-s − 0.353i·8-s + (0.286 − 0.958i)9-s + 1.10i·10-s + 1.06i·11-s + (−0.459 − 0.197i)12-s + (−0.996 + 0.0876i)13-s + (−0.620 + 0.339i)14-s + (1.44 + 0.620i)15-s + (−0.125 + 0.216i)16-s + (0.479 + 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.891 - 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0217084 + 0.0906817i\)
\(L(\frac12)\) \(\approx\) \(0.0217084 + 0.0906817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 1.03i)T \)
7 \( 1 + (-1.37 + 2.26i)T \)
13 \( 1 + (3.59 - 0.316i)T \)
good5 \( 1 + (1.75 + 3.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.53iT - 11T^{2} \)
17 \( 1 + (-1.97 - 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 6.65iT - 19T^{2} \)
23 \( 1 + (-1.11 - 0.642i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.71 - 3.29i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.14 + 4.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.28 - 9.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.84 + 4.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.79 - 4.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0582 + 0.100i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.70 + 3.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.84 - 4.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 - 7.68T + 67T^{2} \)
71 \( 1 + (8.18 + 4.72i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.73 - 2.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.63 + 4.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.03T + 83T^{2} \)
89 \( 1 + (0.177 - 0.307i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.507 + 0.292i)T + (48.5 + 84.0i)T^{2} \)
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

−10.27412939659256279639386145171, −9.502317794082808576386592571877, −8.731633509232160061564882060998, −7.59990593770230695980264421697, −6.99417370957372973046922628596, −5.18619016986600266078801878934, −4.62900066641577523480328455754, −3.74682306024170494410356172782, −1.49622963737799283384043262418, −0.07409787292345103485693082883, 2.04539653380437196431582213952, 3.38927437431671370276743646784, 5.24131342388629173026676562606, 5.91836792610622789186194867021, 7.00146345037808265584585240319, 7.60618801076334934745791442569, 8.313044753775106097650462097456, 9.630603404507824102346952044167, 10.69791181603376762153047907433, 11.19571612410163450900976962798

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