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Properties

Label 2-558-279.103-c1-0-17
Degree $2$
Conductor $558$
Sign $0.992 - 0.123i$
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 + (1.66 − 0.491i)3-s + (−0.978 − 0.207i)4-s − 3.28·5-s + (0.315 + 1.70i)6-s + (−0.188 − 0.581i)7-s + (0.309 − 0.951i)8-s + (2.51 − 1.63i)9-s + (0.343 − 3.26i)10-s + (4.75 + 1.00i)11-s + (−1.72 + 0.135i)12-s + (4.04 − 2.94i)13-s + (0.598 − 0.127i)14-s + (−5.45 + 1.61i)15-s + (0.913 + 0.406i)16-s + (−3.14 − 3.49i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (0.958 − 0.283i)3-s + (−0.489 − 0.103i)4-s − 1.46·5-s + (0.128 + 0.695i)6-s + (−0.0714 − 0.219i)7-s + (0.109 − 0.336i)8-s + (0.838 − 0.544i)9-s + (0.108 − 1.03i)10-s + (1.43 + 0.304i)11-s + (−0.498 + 0.0391i)12-s + (1.12 − 0.815i)13-s + (0.159 − 0.0339i)14-s + (−1.40 + 0.416i)15-s + (0.228 + 0.101i)16-s + (−0.763 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $0.992 - 0.123i$
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: \((2,\ 558,\ (\ :1/2),\ 0.992 - 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59649 + 0.0988148i\)
\(L(\frac12)\) \(\approx\) \(1.59649 + 0.0988148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 0.491i)T \)
31 \( 1 + (4.12 - 3.73i)T \)
good5 \( 1 + 3.28T + 5T^{2} \)
7 \( 1 + (0.188 + 0.581i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-4.75 - 1.00i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-4.04 + 2.94i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.14 + 3.49i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (-4.23 + 1.88i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-5.10 - 5.66i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (0.455 - 4.33i)T + (-28.3 - 6.02i)T^{2} \)
37 \( 1 + (-2.64 + 4.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.55 + 3.30i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.60 + 1.89i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.477 + 0.212i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-0.812 + 0.172i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (5.95 + 2.65i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (4.89 + 8.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 9.21T + 67T^{2} \)
71 \( 1 + (3.94 - 0.839i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-6.22 + 6.91i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (4.89 - 15.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.51 - 2.01i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-0.946 - 2.91i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (11.1 - 12.3i)T + (-10.1 - 96.4i)T^{2} \)
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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

−10.88058615951026156251499065761, −9.362454217528705394900731402250, −8.944958393388738182516571311465, −8.029260551700515294697811804380, −7.18303545320778884235030000670, −6.80124200676185112446796216265, −5.10739921040188430569338414887, −3.83188237812319574691522680302, −3.38545333800107033122216581739, −1.07769105084102611137457802532, 1.42712371396248406400052665804, 3.08493904548996425805055148288, 3.97241844421576129844043987176, 4.38485284134996701726202151418, 6.33997353056868816434939417796, 7.42807877564424993020936134442, 8.562713946178582896233288017468, 8.754439572222693133455385287776, 9.760992778224888493766556389360, 11.00914175692937079287561498971

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