↑

Properties

Label 2-558-279.160-c1-0-16
Degree $2$
Conductor $558$
Sign $0.621 - 0.783i$
Analytic cond. $4.45565$
Root an. cond. $2.11084$
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  + (−0.5 + 0.866i)2-s + (1.68 + 0.419i)3-s + (−0.499 − 0.866i)4-s + (−0.487 + 0.844i)5-s + (−1.20 + 1.24i)6-s + (−0.761 − 1.31i)7-s + 0.999·8-s + (2.64 + 1.40i)9-s + (−0.487 − 0.844i)10-s + 4.79·11-s + (−0.477 − 1.66i)12-s + (3.54 − 6.13i)13-s + 1.52·14-s + (−1.17 + 1.21i)15-s + (−0.5 + 0.866i)16-s + (−0.0792 + 0.137i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.970 + 0.241i)3-s + (−0.249 − 0.433i)4-s + (−0.218 + 0.377i)5-s + (−0.491 + 0.508i)6-s + (−0.287 − 0.498i)7-s + 0.353·8-s + (0.882 + 0.469i)9-s + (−0.154 − 0.267i)10-s + 1.44·11-s + (−0.137 − 0.480i)12-s + (0.981 − 1.70i)13-s + 0.407·14-s + (−0.302 + 0.313i)15-s + (−0.125 + 0.216i)16-s + (−0.0192 + 0.0332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54844 + 0.747508i\)
\(L(\frac12)\) \(\approx\) \(1.54844 + 0.747508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.68 - 0.419i)T \)
31 \( 1 + (4.41 + 3.38i)T \)
good5 \( 1 + (0.487 - 0.844i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.761 + 1.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.79T + 11T^{2} \)
13 \( 1 + (-3.54 + 6.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0792 - 0.137i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.37 - 5.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.16 - 3.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.414 + 0.718i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (3.49 - 6.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.97 - 5.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.52 + 4.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.92 + 8.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.39 + 4.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.94T + 59T^{2} \)
61 \( 1 + (1.80 - 3.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.59 - 2.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.80 - 8.31i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.66 + 11.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.85 + 6.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.24T + 83T^{2} \)
89 \( 1 - 0.134T + 89T^{2} \)
97 \( 1 + (-0.949 + 1.64i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.46208810299983350061723162578, −10.02670493173463896322224556166, −8.907761822096759350063928924199, −8.311148568222911466683238511097, −7.41205301385430142443465430763, −6.60181679800306154841125903742, −5.49082513890479161335843242468, −3.89648843505172885621483524890, −3.40322990697363531328459923789, −1.42731845768238233499451800130, 1.34017453738474605801151899273, 2.49904114490394481564799716170, 3.82826057254776013398988392555, 4.47740993215352939832171576455, 6.47498423091337620359622244265, 7.00723793694126296303586142292, 8.509217125125363841091940872231, 9.079644297189980253315826750633, 9.194094759413972007547277823688, 10.68933123548349688950982698436

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