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Properties

Label 2-558-279.103-c1-0-10
Degree $2$
Conductor $558$
Sign $0.454 - 0.890i$
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 + (−0.768 − 1.55i)3-s + (−0.978 − 0.207i)4-s + 2.90·5-s + (1.62 − 0.601i)6-s + (1.14 + 3.53i)7-s + (0.309 − 0.951i)8-s + (−1.81 + 2.38i)9-s + (−0.304 + 2.89i)10-s + (0.427 + 0.0908i)11-s + (0.428 + 1.67i)12-s + (−3.67 + 2.67i)13-s + (−3.63 + 0.772i)14-s + (−2.23 − 4.51i)15-s + (0.913 + 0.406i)16-s + (0.0519 + 0.0577i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (−0.443 − 0.896i)3-s + (−0.489 − 0.103i)4-s + 1.30·5-s + (0.663 − 0.245i)6-s + (0.434 + 1.33i)7-s + (0.109 − 0.336i)8-s + (−0.606 + 0.794i)9-s + (−0.0961 + 0.914i)10-s + (0.128 + 0.0273i)11-s + (0.123 + 0.484i)12-s + (−1.02 + 0.741i)13-s + (−0.971 + 0.206i)14-s + (−0.576 − 1.16i)15-s + (0.228 + 0.101i)16-s + (0.0126 + 0.0139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.454 - 0.890i$
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: \((2,\ 558,\ (\ :1/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20073 + 0.734983i\)
\(L(\frac12)\) \(\approx\) \(1.20073 + 0.734983i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.768 + 1.55i)T \)
31 \( 1 + (-1.12 + 5.45i)T \)
good5 \( 1 - 2.90T + 5T^{2} \)
7 \( 1 + (-1.14 - 3.53i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.427 - 0.0908i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (3.67 - 2.67i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.0519 - 0.0577i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (-4.84 + 2.15i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-3.11 - 3.45i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (0.787 - 7.49i)T + (-28.3 - 6.02i)T^{2} \)
37 \( 1 + (0.723 - 1.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.36 - 3.89i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.787 + 0.572i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (3.77 + 1.67i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-1.83 + 0.390i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-2.03 - 0.906i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (5.25 + 9.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + (-10.9 + 2.32i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-2.20 + 2.44i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-2.53 + 7.81i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.1 - 4.97i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.32 + 7.15i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (11.1 - 12.4i)T + (-10.1 - 96.4i)T^{2} \)
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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

−11.08560387086600369724685504902, −9.600127071539875588418911640153, −9.260563547246873061840202279128, −8.157285330727669438532760952740, −7.13791019488334720614617354174, −6.38285529320861232529891723755, −5.37747998478645059486211182793, −5.11156653797267366277533683526, −2.64143581486039602200986273659, −1.61776846143351442661718260446, 0.964771866373970445299641495099, 2.65287401291865870335279505051, 3.91953996508155831252692545051, 4.93955276071127576222748822247, 5.65783238081552002225110258825, 6.94301996484060171556134693367, 8.122472289957601129731814918154, 9.418782668521055326379004073393, 9.921852437158771762171002327499, 10.46307588572089733281203371895

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