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Properties

Label 1-548-548.171-r1-0-0
Degree $1$
Conductor $548$
Sign $0.743 + 0.668i$
Analytic cond. $58.8907$
Root an. cond. $58.8907$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.602 − 0.798i)3-s + (−0.982 + 0.183i)5-s + (−0.445 + 0.895i)7-s + (−0.273 − 0.961i)9-s + (−0.932 − 0.361i)11-s + (0.445 − 0.895i)13-s + (−0.445 + 0.895i)15-s + (−0.850 + 0.526i)17-s + (−0.0922 − 0.995i)19-s + (0.445 + 0.895i)21-s + (−0.739 − 0.673i)23-s + (0.932 − 0.361i)25-s + (−0.932 − 0.361i)27-s + (0.739 + 0.673i)29-s + (−0.0922 + 0.995i)31-s + ⋯
L(s)  = 1  + (0.602 − 0.798i)3-s + (−0.982 + 0.183i)5-s + (−0.445 + 0.895i)7-s + (−0.273 − 0.961i)9-s + (−0.932 − 0.361i)11-s + (0.445 − 0.895i)13-s + (−0.445 + 0.895i)15-s + (−0.850 + 0.526i)17-s + (−0.0922 − 0.995i)19-s + (0.445 + 0.895i)21-s + (−0.739 − 0.673i)23-s + (0.932 − 0.361i)25-s + (−0.932 − 0.361i)27-s + (0.739 + 0.673i)29-s + (−0.0922 + 0.995i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(58.8907\)
Root analytic conductor: \(58.8907\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 548,\ (1:\ ),\ 0.743 + 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9526424381 + 0.3655115294i\)
\(L(\frac12)\) \(\approx\) \(0.9526424381 + 0.3655115294i\)
\(L(1)\) \(\approx\) \(0.8759685177 - 0.1420693922i\)
\(L(1)\) \(\approx\) \(0.8759685177 - 0.1420693922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
137 \( 1 \)
good3 \( 1 + (0.602 - 0.798i)T \)
5 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (-0.445 + 0.895i)T \)
11 \( 1 + (-0.932 - 0.361i)T \)
13 \( 1 + (0.445 - 0.895i)T \)
17 \( 1 + (-0.850 + 0.526i)T \)
19 \( 1 + (-0.0922 - 0.995i)T \)
23 \( 1 + (-0.739 - 0.673i)T \)
29 \( 1 + (0.739 + 0.673i)T \)
31 \( 1 + (-0.0922 + 0.995i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (-0.0922 - 0.995i)T \)
47 \( 1 + (0.273 + 0.961i)T \)
53 \( 1 + (0.0922 + 0.995i)T \)
59 \( 1 + (0.273 + 0.961i)T \)
61 \( 1 + (-0.273 + 0.961i)T \)
67 \( 1 + (-0.445 + 0.895i)T \)
71 \( 1 + (-0.932 + 0.361i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.602 + 0.798i)T \)
83 \( 1 + (0.850 + 0.526i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (0.932 + 0.361i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.128966999902648915405089704762, −22.309350096313854820401476893806, −21.14090339310593598252500491261, −20.553074218341751724995932912561, −19.82007638761666800632527463023, −19.17445410563623501662858258664, −18.14440869606410295464881005820, −16.79496524349543666444069886212, −16.12853672899961640384477480736, −15.64220154531302781410740416903, −14.65347435458567609377553496769, −13.703489267317661054352407277911, −13.01017212164542103481783043834, −11.69521942801323849917160731857, −10.921753532228461701183087438341, −9.99991029815584449133397109616, −9.22754005010353158588358594688, −8.02345150336188642917424786377, −7.59267127174845804977050248784, −6.28038279986791758517159223200, −4.75749442251022037351179913392, −4.14652027417152100254939967368, −3.36676964663757991388776021139, −2.09970480263381182558475631657, −0.3082885135699847607841979318, 0.81627831219850789154911840376, 2.55553955777031908972929785106, 2.97017581738347108396386392490, 4.23862662334994155326113333179, 5.66074698776889535750055428003, 6.58266585815891944666409598038, 7.54860852714206365303015850332, 8.47429019226103658903930329646, 8.85303946002063997571495774547, 10.410582956742606544826948720655, 11.2801520251256005744900849231, 12.41655302828216444973862275722, 12.80565907136990769279107173870, 13.78611313474389461873395090120, 14.95526624220735220070594182283, 15.52764275931537398560878874057, 16.18112747724080660553183530456, 17.88914044942487819900523833178, 18.203503715273833713242132614170, 19.19267784658566011175020582344, 19.74318506985682385120914567981, 20.49368076812085336919696176817, 21.65152929555516071716709179418, 22.50893995475170592852253504013, 23.507853618872313110174682087868

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