↑

Properties

Label 2-546-91.76-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.163 - 0.986i$
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  + (0.258 − 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (0.0374 − 0.0374i)5-s + (0.258 + 0.965i)6-s + (−2.54 + 0.733i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.0264 − 0.0458i)10-s + (1.73 + 0.465i)11-s + 12-s + (−2.12 − 2.91i)13-s + (0.0501 + 2.64i)14-s + (−0.0137 + 0.0511i)15-s + (0.500 + 0.866i)16-s + (−3.26 + 5.65i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (0.0167 − 0.0167i)5-s + (0.105 + 0.394i)6-s + (−0.960 + 0.277i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.00837 − 0.0145i)10-s + (0.523 + 0.140i)11-s + 0.288·12-s + (−0.589 − 0.807i)13-s + (0.0133 + 0.706i)14-s + (−0.00353 + 0.0132i)15-s + (0.125 + 0.216i)16-s + (−0.791 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300582 + 0.354453i\)
\(L(\frac12)\) \(\approx\) \(0.300582 + 0.354453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.54 - 0.733i)T \)
13 \( 1 + (2.12 + 2.91i)T \)
good5 \( 1 + (-0.0374 + 0.0374i)T - 5iT^{2} \)
11 \( 1 + (-1.73 - 0.465i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.26 - 5.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 - 5.97i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.84 - 3.94i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.18 + 5.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.81 - 1.81i)T - 31iT^{2} \)
37 \( 1 + (4.19 + 1.12i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.982 - 0.263i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.47 - 5.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.86 + 4.86i)T + 47iT^{2} \)
53 \( 1 + 4.52T + 53T^{2} \)
59 \( 1 + (-9.90 + 2.65i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.84 - 6.87i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.4 - 2.80i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.17 + 3.17i)T + 73iT^{2} \)
79 \( 1 - 5.14T + 79T^{2} \)
83 \( 1 + (6.12 - 6.12i)T - 83iT^{2} \)
89 \( 1 + (-1.38 + 5.15i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.92 + 14.6i)T + (-84.0 + 48.5i)T^{2} \)
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

−11.06684370626100551764384234417, −10.00641749285835068737478280947, −9.770675078994883264854717521060, −8.612790345221859131855531546432, −7.44819739521925669960117269184, −6.07006048098399133425307312763, −5.63717318385467325271812984912, −4.13390099363108099672155364465, −3.41824436712026799761239938866, −1.83151539224426469794365975493, 0.25833799618760689006223099829, 2.55809624993176632405788930989, 4.08790473881301934790277241190, 4.97519781807669875575813256866, 6.22455010328204889190323585986, 6.83415174655169043922941583636, 7.46424282293383098579836580766, 8.935252390875672428690757675108, 9.444680522031357374228710540164, 10.53790872199382634038271339836

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