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Properties

Label 1-588-588.59-r1-0-0
Degree $1$
Conductor $588$
Sign $-0.664 - 0.747i$
Analytic cond. $63.1893$
Root an. cond. $63.1893$
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  + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)11-s + (0.222 + 0.974i)13-s + (0.0747 + 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (0.900 + 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.826 − 0.563i)37-s + (0.623 + 0.781i)41-s + (−0.623 + 0.781i)43-s + (−0.955 − 0.294i)47-s + (−0.826 − 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)5-s + (−0.733 + 0.680i)11-s + (0.222 + 0.974i)13-s + (0.0747 + 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (0.900 + 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.826 − 0.563i)37-s + (0.623 + 0.781i)41-s + (−0.623 + 0.781i)43-s + (−0.955 − 0.294i)47-s + (−0.826 − 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(63.1893\)
Root analytic conductor: \(63.1893\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 588,\ (1:\ ),\ -0.664 - 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0009499887998 + 0.002115432927i\)
\(L(\frac12)\) \(\approx\) \(0.0009499887998 + 0.002115432927i\)
\(L(1)\) \(\approx\) \(0.7260920742 + 0.1354786452i\)
\(L(1)\) \(\approx\) \(0.7260920742 + 0.1354786452i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (0.0747 + 0.997i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (-0.826 - 0.563i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 - T \)
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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

−22.53850533066919272809960505841, −21.58213553616420539012015451764, −20.6966604082205623712267500253, −19.893502147780261993038660678574, −19.204580901916684507614667965799, −18.299994922581563677573107483659, −17.50574459552967460355736381632, −16.32562534910703381564188279150, −15.713325622073138860698689617, −15.10996808082696175857765648333, −13.85758589253460340426432326858, −13.0920596107957604701291434504, −12.17853250421596709306196089929, −11.24257986845660407437055386537, −10.63497073629509011251519771902, −9.39434260501977874426021045677, −8.36744804198974748656499060507, −7.74720441279133849359227925430, −6.76485806726707300970245463016, −5.47547466834873610497598496449, −4.70306933120649297106001575002, −3.44339737633867433102878402483, −2.7289627057715039485749094268, −0.93527245672675316335453661165, −0.000693906817803641982009525238, 1.594398902521471003093171935094, 2.79930411798053213051871091101, 4.0431444149566118276434578690, 4.606044685289335328948023357848, 6.04107519820926677963322170747, 6.94469345283845926429402135332, 7.94562902469431146630351866180, 8.561830212844315661180421878895, 9.84508011194809372756708637440, 10.707601208044250044677247486757, 11.541116300851802539352240767, 12.48378242173172358498024434850, 13.097238738509533689871219174098, 14.65291077203074093180484361713, 14.79982553700059104959965867291, 16.12745470059203604305995538662, 16.498917901183259618482059239294, 17.76397964145932116525601617495, 18.626979838560057137421895968144, 19.26285024870188404561899045225, 20.13729084101335051307817852567, 20.97012922556881669682462706795, 21.78306821051955809799684507518, 22.9093813102256021001731851463, 23.41703047253175995548201700322

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