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Properties

Label 2-558-279.103-c1-0-11
Degree $2$
Conductor $558$
Sign $-0.832 - 0.554i$
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 + (0.100 + 1.72i)3-s + (−0.978 − 0.207i)4-s + 2.97·5-s + (−1.73 − 0.0812i)6-s + (0.990 + 3.04i)7-s + (0.309 − 0.951i)8-s + (−2.97 + 0.346i)9-s + (−0.311 + 2.96i)10-s + (−3.53 − 0.752i)11-s + (0.261 − 1.71i)12-s + (1.13 − 0.824i)13-s + (−3.13 + 0.666i)14-s + (0.298 + 5.15i)15-s + (0.913 + 0.406i)16-s + (4.08 + 4.53i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (0.0577 + 0.998i)3-s + (−0.489 − 0.103i)4-s + 1.33·5-s + (−0.706 − 0.0331i)6-s + (0.374 + 1.15i)7-s + (0.109 − 0.336i)8-s + (−0.993 + 0.115i)9-s + (−0.0984 + 0.936i)10-s + (−1.06 − 0.226i)11-s + (0.0755 − 0.494i)12-s + (0.314 − 0.228i)13-s + (−0.837 + 0.178i)14-s + (0.0769 + 1.33i)15-s + (0.228 + 0.101i)16-s + (0.989 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(558\)    =    \(2 \cdot 3^{2} \cdot 31\)
Sign: $-0.832 - 0.554i$
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: \((2,\ 558,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460966 + 1.52326i\)
\(L(\frac12)\) \(\approx\) \(0.460966 + 1.52326i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.100 - 1.72i)T \)
31 \( 1 + (4.92 - 2.59i)T \)
good5 \( 1 - 2.97T + 5T^{2} \)
7 \( 1 + (-0.990 - 3.04i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (3.53 + 0.752i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-1.13 + 0.824i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.08 - 4.53i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (0.408 - 0.181i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-2.57 - 2.85i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (-0.523 + 4.97i)T + (-28.3 - 6.02i)T^{2} \)
37 \( 1 + (-0.215 + 0.373i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.81 + 2.77i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (7.26 + 5.27i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-9.67 - 4.30i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-6.98 + 1.48i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (8.80 + 3.91i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-1.04 - 1.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (-13.4 + 2.86i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-6.34 + 7.05i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (2.28 - 7.02i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.31 + 3.25i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (1.63 + 5.02i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.09 + 5.65i)T + (-10.1 - 96.4i)T^{2} \)
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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

−10.67461012232462525808376962722, −10.19626169665894962637945634460, −9.257066351761697342167735790851, −8.647174863449032605930012498622, −7.79293992360556498763884623052, −6.10148500132712645277779023752, −5.59173320861820893018089369094, −5.06519851958537261637039304952, −3.43900703922026387050813034315, −2.12751173018824652467200326576, 0.980741905987742802278221372684, 2.09637530728262400670650946913, 3.17086959001407235334574726811, 4.87253498625597690791796795813, 5.71160800949527648917164754244, 6.95573627988019961906497950740, 7.65847085710348880291256120253, 8.725534771644053697040749407917, 9.709667628723822934947282556162, 10.47913955290296378571163311922

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