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Properties

Label 2-558-279.160-c1-0-8
Degree $2$
Conductor $558$
Sign $0.301 - 0.953i$
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 + (−0.735 − 1.56i)3-s + (−0.499 − 0.866i)4-s + (0.112 − 0.194i)5-s + (1.72 + 0.146i)6-s + (1.36 + 2.37i)7-s + 0.999·8-s + (−1.91 + 2.30i)9-s + (0.112 + 0.194i)10-s − 1.78·11-s + (−0.989 + 1.42i)12-s + (−2.36 + 4.08i)13-s − 2.73·14-s + (−0.387 − 0.0329i)15-s + (−0.5 + 0.866i)16-s + (1.76 − 3.06i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.424 − 0.905i)3-s + (−0.249 − 0.433i)4-s + (0.0501 − 0.0868i)5-s + (0.704 + 0.0599i)6-s + (0.517 + 0.896i)7-s + 0.353·8-s + (−0.639 + 0.769i)9-s + (0.0354 + 0.0614i)10-s − 0.539·11-s + (−0.285 + 0.410i)12-s + (−0.654 + 1.13i)13-s − 0.731·14-s + (−0.0999 − 0.00850i)15-s + (−0.125 + 0.216i)16-s + (0.429 − 0.743i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.301 - 0.953i$
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: \((2,\ 558,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723719 + 0.529982i\)
\(L(\frac12)\) \(\approx\) \(0.723719 + 0.529982i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.735 + 1.56i)T \)
31 \( 1 + (0.472 - 5.54i)T \)
good5 \( 1 + (-0.112 + 0.194i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.36 - 2.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
13 \( 1 + (2.36 - 4.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.76 + 3.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.48 + 2.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.28 - 7.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.91 + 5.05i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (5.39 - 9.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 - 3.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.34 - 7.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.95 + 6.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.39 + 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.735T + 59T^{2} \)
61 \( 1 + (-0.690 + 1.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.77 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.76 - 9.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.13 + 1.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.08 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-6.40 + 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.25430214185135264423600229366, −9.840897393448407444296372503733, −9.028189038691084325724842447157, −8.160269549755802047849952948233, −7.26588915329405012013880907005, −6.64823336633767212337972054351, −5.27165893290003282563740860031, −5.06621215245536594910418162429, −2.75863241490234324042271110813, −1.42704798286351533541034424214, 0.66452005209993419296844071447, 2.69709515972558456075231731477, 3.85225725419628714253459701686, 4.79174992218270531807385231584, 5.72520986353285418507490341489, 7.16564101038652420691940116922, 8.109546250130694768269209726776, 8.970009566448183617127422221718, 10.20942010614630142470284487673, 10.49844709899771701388986392580

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