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Properties

Label 1-3724-3724.163-r1-0-0
Degree $1$
Conductor $3724$
Sign $0.934 - 0.356i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (−0.623 − 0.781i)3-s + (−0.988 + 0.149i)5-s + (−0.222 + 0.974i)9-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.733 + 0.680i)15-s + (−0.900 − 0.433i)17-s + (0.900 − 0.433i)23-s + (0.955 − 0.294i)25-s + (0.900 − 0.433i)27-s + (0.826 − 0.563i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)33-s + (0.0747 + 0.997i)37-s + (−0.365 − 0.930i)39-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)3-s + (−0.988 + 0.149i)5-s + (−0.222 + 0.974i)9-s + (−0.955 − 0.294i)11-s + (0.955 + 0.294i)13-s + (0.733 + 0.680i)15-s + (−0.900 − 0.433i)17-s + (0.900 − 0.433i)23-s + (0.955 − 0.294i)25-s + (0.900 − 0.433i)27-s + (0.826 − 0.563i)29-s + (0.5 − 0.866i)31-s + (0.365 + 0.930i)33-s + (0.0747 + 0.997i)37-s + (−0.365 − 0.930i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.934 - 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062579072 - 0.1956489154i\)
\(L(\frac12)\) \(\approx\) \(1.062579072 - 0.1956489154i\)
\(L(1)\) \(\approx\) \(0.6943081994 - 0.1536236042i\)
\(L(1)\) \(\approx\) \(0.6943081994 - 0.1536236042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (0.365 - 0.930i)T \)
43 \( 1 + (0.988 + 0.149i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (-0.623 + 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.826 - 0.563i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.2227719350958694652688530221, −17.91276976722280586261460461481, −17.04612677378196380593257976966, −16.21754568579425255587364517229, −15.77453059980487865154625870610, −15.33437486240944745429438021298, −14.68620533970573792130480449919, −13.59467698538640625296353163390, −12.80194348140514365146415656411, −12.25750930023800289033097787914, −11.40645647799814697549166684470, −10.73669058219906952778807261852, −10.53873875409966358561733630626, −9.35440979151754955840740191924, −8.70133319752694283552446674685, −8.067797539055955541857068010924, −7.11768277513094105923736912481, −6.42613339223748539152082025911, −5.473178049267751268293931608488, −4.85541310870324776633803561521, −4.16947538437520948505432904098, −3.44474106845054039778442094748, −2.69081866506342264146332771320, −1.24932114173798253147225638273, −0.39302027781999775004968819818, 0.49746492876964296721404163869, 1.14138453157511372966107866469, 2.45698472846720850944107349462, 2.948075963832506589784355107630, 4.22843885429557250919250797347, 4.70594114392637229392164454195, 5.7399909911535659266061084734, 6.39961501057699855470560204199, 7.12311868320034951592405586454, 7.79006835449256311041459778049, 8.40265897236147933593384443690, 9.10909389157656683503480421127, 10.48789732504348950434848230153, 10.86294489166469456311695099231, 11.52147039445476964325137863652, 12.09092377047246097965247143061, 12.930560945824691995473187056708, 13.45955203398303798720055537081, 14.094832637214619849896197617515, 15.23864468281023249733514636994, 15.749047971569611297581283140733, 16.290215039025936212738063448154, 17.095871727350423772601960716555, 17.84593307444254912619279201688, 18.546386254222842532578603762754

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