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Properties

Label 2-558-279.103-c1-0-4
Degree $2$
Conductor $558$
Sign $0.163 - 0.986i$
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  + (−0.104 + 0.994i)2-s + (−0.977 − 1.42i)3-s + (−0.978 − 0.207i)4-s − 3.36·5-s + (1.52 − 0.822i)6-s + (−0.555 − 1.71i)7-s + (0.309 − 0.951i)8-s + (−1.08 + 2.79i)9-s + (0.351 − 3.34i)10-s + (0.810 + 0.172i)11-s + (0.658 + 1.60i)12-s + (1.29 − 0.944i)13-s + (1.75 − 0.373i)14-s + (3.28 + 4.80i)15-s + (0.913 + 0.406i)16-s + (4.68 + 5.20i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (−0.564 − 0.825i)3-s + (−0.489 − 0.103i)4-s − 1.50·5-s + (0.622 − 0.335i)6-s + (−0.210 − 0.646i)7-s + (0.109 − 0.336i)8-s + (−0.363 + 0.931i)9-s + (0.111 − 1.05i)10-s + (0.244 + 0.0519i)11-s + (0.190 + 0.462i)12-s + (0.360 − 0.261i)13-s + (0.470 − 0.0999i)14-s + (0.848 + 1.24i)15-s + (0.228 + 0.101i)16-s + (1.13 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(4.45565\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{558} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 558,\ (\ :1/2),\ 0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441601 + 0.374368i\)
\(L(\frac12)\) \(\approx\) \(0.441601 + 0.374368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (0.977 + 1.42i)T \)
31 \( 1 + (-0.974 + 5.48i)T \)
good5 \( 1 + 3.36T + 5T^{2} \)
7 \( 1 + (0.555 + 1.71i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.810 - 0.172i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-1.29 + 0.944i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.68 - 5.20i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (3.53 - 1.57i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-1.28 - 1.43i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (0.934 - 8.88i)T + (-28.3 - 6.02i)T^{2} \)
37 \( 1 + (1.40 - 2.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.953 - 0.692i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.27 - 4.55i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.65 - 0.736i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (7.23 - 1.53i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-2.05 - 0.915i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-5.59 - 9.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + (2.45 - 0.521i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-4.29 + 4.76i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-2.15 + 6.64i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.67 - 2.08i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-0.847 - 2.60i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.13 + 2.36i)T + (-10.1 - 96.4i)T^{2} \)
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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

−10.95443857847036434671363814424, −10.34289248407242988188513679342, −8.781473634035921425983969961317, −7.923928992782788214898076089388, −7.50429701265103863448710747266, −6.57665234905861444603515334037, −5.68000412230900878916227839678, −4.37840842803937710591837666894, −3.48327715448044930904216633436, −1.11679103642725203018260939396, 0.46053853054858335354159599643, 2.88565671611529928175416140842, 3.85139013629021838181821236059, 4.62506594730990476633747730306, 5.71268783755739217201763349217, 6.99319110685609159166440898025, 8.169632540190595543218446175204, 8.996421013866546055994738217559, 9.755422866858738600311873728188, 10.83514533129774764250030971315

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