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Properties

Label 2-558-279.185-c1-0-18
Degree $2$
Conductor $558$
Sign $0.497 + 0.867i$
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  + (0.866 − 0.5i)2-s + (−1.37 + 1.05i)3-s + (0.499 − 0.866i)4-s + (1.74 + 1.00i)5-s + (−0.665 + 1.59i)6-s + (−1.62 − 2.80i)7-s − 0.999i·8-s + (0.787 − 2.89i)9-s + 2.01·10-s + (−2.86 − 4.95i)11-s + (0.222 + 1.71i)12-s + (2.00 + 1.15i)13-s + (−2.80 − 1.62i)14-s + (−3.46 + 0.449i)15-s + (−0.5 − 0.866i)16-s + 4.78·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.794 + 0.607i)3-s + (0.249 − 0.433i)4-s + (0.782 + 0.451i)5-s + (−0.271 + 0.652i)6-s + (−0.612 − 1.06i)7-s − 0.353i·8-s + (0.262 − 0.964i)9-s + 0.638·10-s + (−0.862 − 1.49i)11-s + (0.0643 + 0.495i)12-s + (0.556 + 0.321i)13-s + (−0.750 − 0.433i)14-s + (−0.895 + 0.116i)15-s + (−0.125 − 0.216i)16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42074 - 0.823528i\)
\(L(\frac12)\) \(\approx\) \(1.42074 - 0.823528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (1.37 - 1.05i)T \)
31 \( 1 + (2.35 - 5.04i)T \)
good5 \( 1 + (-1.74 - 1.00i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.62 + 2.80i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 + 4.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.00 - 1.15i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.78T + 17T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 + (-0.788 + 1.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.991 + 1.71i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + 3.80iT - 37T^{2} \)
41 \( 1 + (-6.80 - 3.92i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.65 + 2.11i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.49 - 0.862i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + (12.7 + 7.34i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.68 + 2.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.23 + 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.99iT - 71T^{2} \)
73 \( 1 - 3.40iT - 73T^{2} \)
79 \( 1 + (-8.21 + 4.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.74 - 11.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (-7.16 - 12.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82684962136048186873242160808, −10.00666872987425297363720873794, −9.366002027043183637736158651664, −7.76768562381341695869769671215, −6.55385544412013229730028631505, −5.90553826683184032465563103491, −5.13958375005436166912919679376, −3.73466086524673057916063624003, −3.08324270226030852660593643676, −0.911462921358602764698363263136, 1.71493058292788762173314672645, 2.95771427296444004464436619472, 4.78118696140062751633164728021, 5.63515934614337089053011566898, 5.92094609532909297198495467795, 7.25376786838439853012967139653, 7.87296456539551709416473812101, 9.327255316669810648394185768331, 9.946000856880508053987824864432, 11.11870811303880570709199870779

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