↑

Properties

Label 2-546-91.4-c1-0-8
Degree $2$
Conductor $546$
Sign $0.999 - 0.0386i$
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 + (0.5 + 0.866i)3-s − 4-s + (0.620 − 0.358i)5-s + (0.866 − 0.5i)6-s + (−1.94 + 1.78i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (−0.358 − 0.620i)10-s + (2.22 − 1.28i)11-s + (−0.5 − 0.866i)12-s + (3.57 + 0.467i)13-s + (1.78 + 1.94i)14-s + (0.620 + 0.358i)15-s + 16-s + 0.0248·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 + 0.499i)3-s − 0.5·4-s + (0.277 − 0.160i)5-s + (0.353 − 0.204i)6-s + (−0.736 + 0.676i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.113 − 0.196i)10-s + (0.671 − 0.387i)11-s + (−0.144 − 0.249i)12-s + (0.991 + 0.129i)13-s + (0.478 + 0.520i)14-s + (0.160 + 0.0924i)15-s + 0.250·16-s + 0.00602·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0386i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.999 - 0.0386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57110 + 0.0303569i\)
\(L(\frac12)\) \(\approx\) \(1.57110 + 0.0303569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.94 - 1.78i)T \)
13 \( 1 + (-3.57 - 0.467i)T \)
good5 \( 1 + (-0.620 + 0.358i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.22 + 1.28i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.0248T + 17T^{2} \)
19 \( 1 + (-5.98 - 3.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.39T + 23T^{2} \)
29 \( 1 + (2.77 - 4.80i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.92 + 2.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.17iT - 37T^{2} \)
41 \( 1 + (-4.54 - 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.37 + 11.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.32 + 2.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.72 - 8.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.25iT - 59T^{2} \)
61 \( 1 + (-0.326 + 0.564i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.62 + 0.938i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.52 + 4.34i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.7 + 7.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.194 + 0.336i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.85iT - 83T^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 + (1.78 - 1.03i)T + (48.5 - 84.0i)T^{2} \)
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

−10.88054517778113520128209161542, −9.777328800180447250460404902351, −9.149516937972364199183079250994, −8.680141112961920844794961907212, −7.25494196038135618771308309354, −5.93577115358652804069772852544, −5.19606110306554392116870756901, −3.67480499766173242387914556517, −3.12950531077647994483018014094, −1.47821720146009888914686830739, 1.07116278357664680618666519713, 3.01534754416965105059576603767, 4.06511457477462009139631361609, 5.44355176594182724905557450759, 6.51921142902262717649113779655, 7.04020136387590709688741980498, 7.915297673447350782255018432723, 9.147734835544283300494483962191, 9.528072301702039804966747152965, 10.74973260657412485754130476031

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