↑

Properties

Label 2-546-21.20-c1-0-14
Degree $2$
Conductor $546$
Sign $0.164 + 0.986i$
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  − i·2-s + (−1.72 − 0.146i)3-s − 4-s + 3.83·5-s + (−0.146 + 1.72i)6-s + (−0.655 − 2.56i)7-s + i·8-s + (2.95 + 0.507i)9-s − 3.83i·10-s + 3.53i·11-s + (1.72 + 0.146i)12-s − i·13-s + (−2.56 + 0.655i)14-s + (−6.61 − 0.563i)15-s + 16-s + 6.47·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.996 − 0.0848i)3-s − 0.5·4-s + 1.71·5-s + (−0.0600 + 0.704i)6-s + (−0.247 − 0.968i)7-s + 0.353i·8-s + (0.985 + 0.169i)9-s − 1.21i·10-s + 1.06i·11-s + (0.498 + 0.0424i)12-s − 0.277i·13-s + (−0.685 + 0.175i)14-s + (−1.70 − 0.145i)15-s + 0.250·16-s + 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03484 - 0.876562i\)
\(L(\frac12)\) \(\approx\) \(1.03484 - 0.876562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.72 + 0.146i)T \)
7 \( 1 + (0.655 + 2.56i)T \)
13 \( 1 + iT \)
good5 \( 1 - 3.83T + 5T^{2} \)
11 \( 1 - 3.53iT - 11T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 + 1.55iT - 19T^{2} \)
23 \( 1 - 3.87iT - 23T^{2} \)
29 \( 1 + 6.70iT - 29T^{2} \)
31 \( 1 + 9.76iT - 31T^{2} \)
37 \( 1 + 0.597T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 0.604T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + 6.84iT - 53T^{2} \)
59 \( 1 - 1.74T + 59T^{2} \)
61 \( 1 + 5.28iT - 61T^{2} \)
67 \( 1 - 5.39T + 67T^{2} \)
71 \( 1 - 9.18iT - 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 + 3.29T + 79T^{2} \)
83 \( 1 - 2.06T + 83T^{2} \)
89 \( 1 - 1.46T + 89T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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

−10.35706146554417366388760131969, −9.898825009336788034376739100558, −9.573496594280620093300065589635, −7.77235238185719425982911300095, −6.82131414730190407024657751121, −5.79208473408837854839061196111, −5.11438573548834529226198986465, −3.90188702584996152548964280511, −2.21526915121517962056841104945, −1.06959639148720620644929119079, 1.43286967884311881044945785822, 3.18153150445963728054033258647, 5.07271356828813662772886116360, 5.59906927007438110467410430996, 6.17653491524693895856032011696, 6.98029885053193266867767010436, 8.534359668297378299089505035499, 9.218147940675302709522120470046, 10.15227971219411276827012018255, 10.67434023468377834338371619950

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