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Properties

Label 2-558-279.160-c1-0-24
Degree $2$
Conductor $558$
Sign $0.778 + 0.627i$
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 + (1.72 − 0.199i)3-s + (−0.499 − 0.866i)4-s + (0.513 − 0.889i)5-s + (−0.687 + 1.58i)6-s + (−2.21 − 3.84i)7-s + 0.999·8-s + (2.92 − 0.687i)9-s + (0.513 + 0.889i)10-s + 0.288·11-s + (−1.03 − 1.39i)12-s + (−2.84 + 4.92i)13-s + 4.43·14-s + (0.705 − 1.63i)15-s + (−0.5 + 0.866i)16-s + (3.27 − 5.67i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.993 − 0.115i)3-s + (−0.249 − 0.433i)4-s + (0.229 − 0.397i)5-s + (−0.280 + 0.649i)6-s + (−0.838 − 1.45i)7-s + 0.353·8-s + (0.973 − 0.229i)9-s + (0.162 + 0.281i)10-s + 0.0870·11-s + (−0.298 − 0.401i)12-s + (−0.788 + 1.36i)13-s + 1.18·14-s + (0.182 − 0.421i)15-s + (−0.125 + 0.216i)16-s + (0.795 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44558 - 0.509961i\)
\(L(\frac12)\) \(\approx\) \(1.44558 - 0.509961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.199i)T \)
31 \( 1 + (-5.46 + 1.05i)T \)
good5 \( 1 + (-0.513 + 0.889i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.21 + 3.84i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.288T + 11T^{2} \)
13 \( 1 + (2.84 - 4.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.27 + 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.05 + 7.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.342 + 0.593i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-0.516 + 0.894i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.50 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.687 + 1.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.41 - 4.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.03 + 3.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.82T + 59T^{2} \)
61 \( 1 + (2.40 - 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.37 - 4.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.62 - 6.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.25 - 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.06 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.65T + 83T^{2} \)
89 \( 1 - 5.37T + 89T^{2} \)
97 \( 1 + (3.01 - 5.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.04568363992219966077728291181, −9.749564133433044720625490235284, −8.999291648962852937585758184841, −7.915542337161162525122908576438, −7.00225664128878754581166054071, −6.70835579520528761362495456315, −4.89124589561736257910380198229, −4.07857126043264459588663337600, −2.69461478792230556466723810551, −0.928814262979280024201263924991, 1.90522383945684676777111228082, 2.97469302374402092521554947188, 3.56373659381490930058833947186, 5.31546145185676423531041784329, 6.27828547756468760532538713293, 7.74911418838067140689559441167, 8.266391629903557417089281511264, 9.271683920926881748189683799553, 10.05499469500639348685531537397, 10.33196064649125306946646910006

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