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Properties

Label 2-558-279.160-c1-0-5
Degree $2$
Conductor $558$
Sign $-0.723 - 0.690i$
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 + (1.21 − 1.23i)3-s + (−0.499 − 0.866i)4-s + (−2.00 + 3.46i)5-s + (0.465 + 1.66i)6-s + (−0.535 − 0.927i)7-s + 0.999·8-s + (−0.0635 − 2.99i)9-s + (−2.00 − 3.46i)10-s + 0.881·11-s + (−1.67 − 0.430i)12-s + (−2.83 + 4.91i)13-s + 1.07·14-s + (1.86 + 6.68i)15-s + (−0.5 + 0.866i)16-s + (−2.98 + 5.17i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.699 − 0.714i)3-s + (−0.249 − 0.433i)4-s + (−0.895 + 1.55i)5-s + (0.190 + 0.681i)6-s + (−0.202 − 0.350i)7-s + 0.353·8-s + (−0.0211 − 0.999i)9-s + (−0.633 − 1.09i)10-s + 0.265·11-s + (−0.484 − 0.124i)12-s + (−0.787 + 1.36i)13-s + 0.286·14-s + (0.481 + 1.72i)15-s + (−0.125 + 0.216i)16-s + (−0.724 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $-0.723 - 0.690i$
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: \((2,\ 558,\ (\ :1/2),\ -0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306026 + 0.763364i\)
\(L(\frac12)\) \(\approx\) \(0.306026 + 0.763364i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.21 + 1.23i)T \)
31 \( 1 + (1.72 + 5.29i)T \)
good5 \( 1 + (2.00 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.535 + 0.927i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.881T + 11T^{2} \)
13 \( 1 + (2.83 - 4.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.98 - 5.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.421 - 0.730i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.62 - 6.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.05 - 1.82i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (3.92 - 6.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.03 + 1.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.99 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.23 - 7.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.79 + 4.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.19T + 59T^{2} \)
61 \( 1 + (4.62 - 8.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.12 + 3.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.16 + 3.74i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.63 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.53 + 4.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + 5.92T + 89T^{2} \)
97 \( 1 + (-8.34 + 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.16589673780773443885999131268, −10.07055058837710731076907139930, −9.208740817607269679562089490132, −8.197127476225634373286269091892, −7.33242068367553646985749734155, −6.90778778536463769422100413051, −6.20509530574221794161578224129, −4.22641860700353160220021759867, −3.35744163307474399439320825280, −1.95827976991151643616132800313, 0.47506402266723580973927480799, 2.46078939217552833074141495748, 3.58252509420954659722442858276, 4.71390960425199308558499795230, 5.17781050639939803854749455046, 7.29709355789176979444199976862, 8.140547192106956525556092196206, 8.926413777088784473680434532763, 9.259631165215089121890930052885, 10.35842556715483146454525627053

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