↑

Properties

Label 2-546-91.89-c1-0-10
Degree $2$
Conductor $546$
Sign $0.355 - 0.934i$
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  + (0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s + 1.00i·4-s + (0.0524 + 0.195i)5-s + (0.258 + 0.965i)6-s + (1.42 − 2.22i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.101 + 0.175i)10-s + (0.723 + 2.70i)11-s + (−0.500 + 0.866i)12-s + (1.90 + 3.06i)13-s + (2.58 − 0.564i)14-s + (−0.0524 + 0.195i)15-s − 1.00·16-s − 2.69·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s + 0.500i·4-s + (0.0234 + 0.0874i)5-s + (0.105 + 0.394i)6-s + (0.539 − 0.841i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (−0.0320 + 0.0554i)10-s + (0.218 + 0.814i)11-s + (−0.144 + 0.250i)12-s + (0.528 + 0.848i)13-s + (0.690 − 0.150i)14-s + (−0.0135 + 0.0505i)15-s − 0.250·16-s − 0.652·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.355 - 0.934i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.355 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92902 + 1.33004i\)
\(L(\frac12)\) \(\approx\) \(1.92902 + 1.33004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-1.42 + 2.22i)T \)
13 \( 1 + (-1.90 - 3.06i)T \)
good5 \( 1 + (-0.0524 - 0.195i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.723 - 2.70i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 + (-3.88 - 1.04i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.27iT - 23T^{2} \)
29 \( 1 + (-1.82 - 3.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.67 + 1.51i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.16 - 1.16i)T - 37iT^{2} \)
41 \( 1 + (10.6 + 2.85i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.73 + 3.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.83 + 1.56i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.577 - 1.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.48 + 3.48i)T + 59iT^{2} \)
61 \( 1 + (-1.74 + 1.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.87 + 2.64i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.05 + 0.283i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.07 + 11.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0814 - 0.141i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.03 - 5.03i)T - 83iT^{2} \)
89 \( 1 + (10.7 + 10.7i)T + 89iT^{2} \)
97 \( 1 + (2.70 + 10.0i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.95263237259383578543969315962, −10.09774663346450374512611372282, −9.036917746674054908966339645441, −8.275299470215122835788203654196, −7.17341640983247419767459848350, −6.69275202160444440178552434749, −5.14098684129710754966907572216, −4.36056488127067078366918926023, −3.46640118334841142580603627443, −1.85436093728896888497997613335, 1.33799371758316384110948896725, 2.72214912776793081529192341886, 3.60381804690706030963214399227, 5.05293803679581876473058517751, 5.79068410628331811398712278223, 6.94492552143343787802632634586, 8.208748848214910113281358784233, 8.801595648579382917271757965471, 9.715925207173041964070158422520, 10.92508865004125546360057843326

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