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Properties

Label 2-558-279.211-c1-0-9
Degree $2$
Conductor $558$
Sign $0.698 + 0.715i$
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 + (−1.35 − 1.08i)3-s + (−0.499 + 0.866i)4-s + (0.281 + 0.487i)5-s + (−0.260 + 1.71i)6-s + (0.502 − 0.870i)7-s + 0.999·8-s + (0.660 + 2.92i)9-s + (0.281 − 0.487i)10-s − 0.144·11-s + (1.61 − 0.630i)12-s + (2.35 + 4.08i)13-s − 1.00·14-s + (0.146 − 0.963i)15-s + (−0.5 − 0.866i)16-s + (0.900 + 1.55i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.781 − 0.624i)3-s + (−0.249 + 0.433i)4-s + (0.125 + 0.217i)5-s + (−0.106 + 0.699i)6-s + (0.189 − 0.328i)7-s + 0.353·8-s + (0.220 + 0.975i)9-s + (0.0889 − 0.154i)10-s − 0.0435·11-s + (0.465 − 0.182i)12-s + (0.654 + 1.13i)13-s − 0.268·14-s + (0.0378 − 0.248i)15-s + (−0.125 − 0.216i)16-s + (0.218 + 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.698 + 0.715i$
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: \((2,\ 558,\ (\ :1/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897101 - 0.378149i\)
\(L(\frac12)\) \(\approx\) \(0.897101 - 0.378149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.08i)T \)
31 \( 1 + (-4.75 - 2.90i)T \)
good5 \( 1 + (-0.281 - 0.487i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.502 + 0.870i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.144T + 11T^{2} \)
13 \( 1 + (-2.35 - 4.08i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.900 - 1.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.19 - 2.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.87 + 3.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.145 + 0.251i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-1.95 - 3.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.763 + 1.32i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.55 + 7.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.11 + 8.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.18 + 3.77i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 + (-1.68 - 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.87 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.93 + 5.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.10 + 3.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.92 - 5.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.17T + 83T^{2} \)
89 \( 1 + 6.44T + 89T^{2} \)
97 \( 1 + (-2.71 - 4.69i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.63464988199907963241609859300, −10.17140434679075674403219259789, −8.861845838794052549293234972319, −8.055828920124165690604855972550, −6.96944388610797112271255434272, −6.28491157869578406698597461179, −4.99410934497910983288968189906, −3.91353911065928250285133925564, −2.31350414369264506158853156274, −1.08118862682528103481473692065, 0.956568894775976311929072018853, 3.20025074803121119182023251045, 4.63150873699592568523795405915, 5.45434536967467855967235821706, 6.11575736342298120472709000849, 7.26665844340274362421845409294, 8.258052535271227436981147363391, 9.256486325323203283932842304694, 9.836538171213809447643534866223, 10.91118578845706888063268039522

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