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Properties

Label 2-558-279.103-c1-0-12
Degree $2$
Conductor $558$
Sign $0.987 + 0.154i$
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  + (0.104 − 0.994i)2-s + (−1.31 + 1.12i)3-s + (−0.978 − 0.207i)4-s + 3.38·5-s + (0.983 + 1.42i)6-s + (0.241 + 0.743i)7-s + (−0.309 + 0.951i)8-s + (0.460 − 2.96i)9-s + (0.353 − 3.36i)10-s + (−0.400 − 0.0851i)11-s + (1.52 − 0.828i)12-s + (0.305 − 0.222i)13-s + (0.764 − 0.162i)14-s + (−4.45 + 3.81i)15-s + (0.913 + 0.406i)16-s + (2.31 + 2.57i)17-s + ⋯
L(s)  = 1  + (0.0739 − 0.703i)2-s + (−0.759 + 0.650i)3-s + (−0.489 − 0.103i)4-s + 1.51·5-s + (0.401 + 0.582i)6-s + (0.0913 + 0.281i)7-s + (−0.109 + 0.336i)8-s + (0.153 − 0.988i)9-s + (0.111 − 1.06i)10-s + (−0.120 − 0.0256i)11-s + (0.439 − 0.239i)12-s + (0.0847 − 0.0616i)13-s + (0.204 − 0.0434i)14-s + (−1.14 + 0.984i)15-s + (0.228 + 0.101i)16-s + (0.561 + 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.987 + 0.154i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48069 - 0.115090i\)
\(L(\frac12)\) \(\approx\) \(1.48069 - 0.115090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (1.31 - 1.12i)T \)
31 \( 1 + (-5.16 - 2.08i)T \)
good5 \( 1 - 3.38T + 5T^{2} \)
7 \( 1 + (-0.241 - 0.743i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.400 + 0.0851i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.305 + 0.222i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.31 - 2.57i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (0.142 - 0.0635i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-4.18 - 4.65i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (-0.344 + 3.27i)T + (-28.3 - 6.02i)T^{2} \)
37 \( 1 + (-2.57 + 4.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.69 + 2.68i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.627 - 0.455i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.313 + 0.139i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-4.56 + 0.969i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (6.69 + 2.97i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-5.59 - 9.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + (9.81 - 2.08i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (3.22 - 3.58i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (2.36 - 7.26i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.1 - 4.98i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (0.397 + 1.22i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.49 - 3.88i)T + (-10.1 - 96.4i)T^{2} \)
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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

−10.62684630584354433983403419332, −9.973847597273226674930628177884, −9.425213531333412888832511676812, −8.490241437336408188521472741748, −6.83140376815512066877205878213, −5.68702379935658676329943487089, −5.39403997507300744135850723175, −4.08115733812270686240940957114, −2.75251938120493781482846863261, −1.34511515380378410178367020310, 1.15987462474118960808516617172, 2.65609234618630504785945852713, 4.69535373812338918843058682870, 5.42075620371394973465888149805, 6.29585438710900251657762053259, 6.89001250932969523831790007500, 7.918480077984335856107930056299, 8.995978517389533734158385900056, 9.980799906385989561887386073884, 10.61538746960734375529912677970

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