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Properties

Label 2-558-279.211-c1-0-1
Degree $2$
Conductor $558$
Sign $0.495 - 0.868i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.59 + 0.686i)3-s + (−0.499 + 0.866i)4-s + (−1.38 − 2.39i)5-s + (1.38 + 1.03i)6-s + (−0.281 + 0.486i)7-s + 0.999·8-s + (2.05 − 2.18i)9-s + (−1.38 + 2.39i)10-s − 3.99·11-s + (0.200 − 1.72i)12-s + (0.735 + 1.27i)13-s + 0.562·14-s + (3.83 + 2.85i)15-s + (−0.5 − 0.866i)16-s + (−1.03 − 1.79i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.918 + 0.396i)3-s + (−0.249 + 0.433i)4-s + (−0.617 − 1.06i)5-s + (0.567 + 0.422i)6-s + (−0.106 + 0.184i)7-s + 0.353·8-s + (0.685 − 0.727i)9-s + (−0.436 + 0.756i)10-s − 1.20·11-s + (0.0578 − 0.496i)12-s + (0.203 + 0.353i)13-s + 0.150·14-s + (0.990 + 0.737i)15-s + (−0.125 − 0.216i)16-s + (−0.250 − 0.434i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.495 - 0.868i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.495 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335698 + 0.195065i\)
\(L(\frac12)\) \(\approx\) \(0.335698 + 0.195065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.686i)T \)
31 \( 1 + (-3.62 - 4.22i)T \)
good5 \( 1 + (1.38 + 2.39i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.281 - 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.99T + 11T^{2} \)
13 \( 1 + (-0.735 - 1.27i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.03 + 1.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.45 - 2.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.12 - 1.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 3.71i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.11 - 5.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.87 - 10.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.96 - 5.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.731 + 1.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 + (6.92 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.84 + 4.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.14 + 3.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.67 - 8.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.516 + 0.893i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.04T + 83T^{2} \)
89 \( 1 + 3.63T + 89T^{2} \)
97 \( 1 + (7.94 + 13.7i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.00911797131742532333407057487, −10.08853211842496926625126554730, −9.348878233612188082788358324675, −8.385994585448825156151061992698, −7.57497679490588003826473270112, −6.21439688185833225911976216402, −5.00759156492057219822574247331, −4.49391891844076121439616036526, −3.13323557114480774675134354203, −1.19632832323682388073603358247, 0.32255010376207994991077226328, 2.50096995929209979895146144454, 4.09668355312736396897148601398, 5.33375778194722458244223221225, 6.18683282767565671276328467950, 7.13967441328228438077072257152, 7.59970387083054732792183110276, 8.569761251444293575639435908393, 10.15695558355490288552827908361, 10.51422449750675025239771403278

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