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Properties

Label 2-558-279.67-c1-0-8
Degree $2$
Conductor $558$
Sign $0.114 - 0.993i$
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 + (0.518 + 1.65i)3-s + (−0.499 + 0.866i)4-s + 2.34·5-s + (1.17 − 1.27i)6-s − 3.04·7-s + 0.999·8-s + (−2.46 + 1.71i)9-s + (−1.17 − 2.03i)10-s + (−2.92 + 5.05i)11-s + (−1.69 − 0.377i)12-s + 5.40·13-s + (1.52 + 2.63i)14-s + (1.21 + 3.87i)15-s + (−0.5 − 0.866i)16-s + (1.83 − 3.18i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.299 + 0.954i)3-s + (−0.249 + 0.433i)4-s + 1.04·5-s + (0.478 − 0.520i)6-s − 1.14·7-s + 0.353·8-s + (−0.821 + 0.570i)9-s + (−0.370 − 0.642i)10-s + (−0.880 + 1.52i)11-s + (−0.487 − 0.109i)12-s + 1.49·13-s + (0.406 + 0.703i)14-s + (0.313 + 1.00i)15-s + (−0.125 − 0.216i)16-s + (0.445 − 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.114 - 0.993i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873446 + 0.778842i\)
\(L(\frac12)\) \(\approx\) \(0.873446 + 0.778842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.518 - 1.65i)T \)
31 \( 1 + (-4.89 - 2.64i)T \)
good5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 + 3.04T + 7T^{2} \)
11 \( 1 + (2.92 - 5.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.40T + 13T^{2} \)
17 \( 1 + (-1.83 + 3.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.58 - 4.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.63 - 4.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.623 + 1.07i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (2.03 - 3.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.94T + 41T^{2} \)
43 \( 1 - 9.83T + 43T^{2} \)
47 \( 1 + (4.27 + 7.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.39 - 9.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.38 + 4.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.03 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + (3.74 + 6.48i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.09 + 3.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 2.67T + 79T^{2} \)
83 \( 1 + (-4.38 + 7.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.52T + 89T^{2} \)
97 \( 1 + (1.79 + 3.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.42094372522891112691430008840, −10.09873765072608043587490901580, −9.570401257578922145429888101726, −8.705762005090699963464927275193, −7.62415961668885034940537270449, −6.24570088321528357002212044958, −5.35802877908059632180174115982, −4.08320209107256708672952902503, −3.08380320820157127337534550369, −1.96348494178833840374270644263, 0.71588678923561594507140575579, 2.38855913055960886168178976504, 3.55466561151250064199034786361, 5.64620830706607838573602752679, 6.16596635495386021696754352978, 6.65042764474461854536464638334, 8.118171988666854421866644360865, 8.579337793645534889775220614806, 9.457844500552517154727152869785, 10.47314918833520619618416894935

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