↑

Properties

Label 2-546-91.74-c1-0-10
Degree $2$
Conductor $546$
Sign $0.513 + 0.858i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.651 + 1.12i)5-s + (0.5 + 0.866i)6-s + (2.21 − 1.45i)7-s − 8-s + (−0.499 + 0.866i)9-s + (−0.651 − 1.12i)10-s + (−3.04 − 5.27i)11-s + (−0.5 − 0.866i)12-s + (2.61 + 2.47i)13-s + (−2.21 + 1.45i)14-s + (0.651 − 1.12i)15-s + 16-s + 3.48·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (0.291 + 0.504i)5-s + (0.204 + 0.353i)6-s + (0.835 − 0.548i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.205 − 0.356i)10-s + (−0.918 − 1.59i)11-s + (−0.144 − 0.249i)12-s + (0.726 + 0.687i)13-s + (−0.591 + 0.388i)14-s + (0.168 − 0.291i)15-s + 0.250·16-s + 0.845·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.513 + 0.858i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.513 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908594 - 0.515262i\)
\(L(\frac12)\) \(\approx\) \(0.908594 - 0.515262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.21 + 1.45i)T \)
13 \( 1 + (-2.61 - 2.47i)T \)
good5 \( 1 + (-0.651 - 1.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.04 + 5.27i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.48T + 17T^{2} \)
19 \( 1 + (0.835 - 1.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.816T + 23T^{2} \)
29 \( 1 + (0.243 - 0.421i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.256 + 0.444i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + (-4.81 + 8.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.00 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.57 + 11.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.24 - 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.15T + 59T^{2} \)
61 \( 1 + (-1.11 + 1.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.07 - 1.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.44 - 9.42i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.44 + 4.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.95 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.486T + 83T^{2} \)
89 \( 1 + 7.69T + 89T^{2} \)
97 \( 1 + (7.84 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.79961110308251192661518361507, −9.979476936583977675478037990678, −8.571409931530944805230620439590, −8.119255798407416851210188561424, −7.15489851744854712839597210547, −6.18276291904024301027994894549, −5.38508167140351165301309916725, −3.70356358094886196962568114878, −2.31992689995320013161384583802, −0.890054047741670988296416720061, 1.43240613802814367011606536744, 2.82349146630522599430417484883, 4.60546563391615815983994876225, 5.27970429958885825571807354038, 6.31911558753334945454375636570, 7.78681004594213933498925707246, 8.166171441584907334888718917362, 9.466697228009368687094283078314, 9.828327778749228475666766435289, 10.90737697246881909816889915052

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