↑

Properties

Label 1-3724-3724.239-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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.365 + 0.930i)3-s + (0.365 − 0.930i)5-s + (−0.733 − 0.680i)9-s + (0.222 + 0.974i)11-s + (0.955 + 0.294i)13-s + (0.733 + 0.680i)15-s + (0.0747 + 0.997i)17-s + (−0.0747 + 0.997i)23-s + (−0.733 − 0.680i)25-s + (0.900 − 0.433i)27-s + (0.826 − 0.563i)29-s − 31-s + (−0.988 − 0.149i)33-s + (−0.900 − 0.433i)37-s + (−0.623 + 0.781i)39-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)3-s + (0.365 − 0.930i)5-s + (−0.733 − 0.680i)9-s + (0.222 + 0.974i)11-s + (0.955 + 0.294i)13-s + (0.733 + 0.680i)15-s + (0.0747 + 0.997i)17-s + (−0.0747 + 0.997i)23-s + (−0.733 − 0.680i)25-s + (0.900 − 0.433i)27-s + (0.826 − 0.563i)29-s − 31-s + (−0.988 − 0.149i)33-s + (−0.900 − 0.433i)37-s + (−0.623 + 0.781i)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: $\chi_{3724} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.934 - 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.841383228 - 0.3390473621i\)
\(L(\frac12)\) \(\approx\) \(1.841383228 - 0.3390473621i\)
\(L(1)\) \(\approx\) \(1.015226418 + 0.1536019513i\)
\(L(1)\) \(\approx\) \(1.015226418 + 0.1536019513i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−18.51924779239854097729801067194, −17.92661182727838158099633130472, −17.38817886975769205425533817948, −16.27591189109509972832441591227, −16.11309863613048084502880539872, −14.82154416852387929153756577756, −14.215873104277622094917329039452, −13.69923051750939541193493431214, −13.10854155578479750041850590269, −12.25791766142717387167254522886, −11.46625207228590086548468025709, −10.948621721883841996313342356997, −10.43360482143568316973041762300, −9.35175774138066845556067021595, −8.516781781125815175943042262194, −7.919768360262753359789615817349, −6.89818827026583663643398016652, −6.60839253723044170490070679171, −5.78021509576052861899353399926, −5.24671725840725544885795919079, −3.94822483409578139164865586730, −2.99105593655389748021777967267, −2.552810645632612596202448120792, −1.405698258787118014523068717413, −0.72788393361454187075276735652, 0.385462149663909202180375344231, 1.449190466196105242385680412666, 2.14487621161550573996407194398, 3.603660742247972619705365512386, 3.98066658520213290173451680570, 4.81916493559627654666739941744, 5.50623359747469603793618486121, 6.105156580154573224472189161733, 6.9714490873392929957729566970, 8.12152309736083487995754827312, 8.73443289418193329689998840969, 9.428980945907847678509392403771, 9.93516978865543725044268842113, 10.72605868309955560375478509355, 11.437777085509057642193931942557, 12.23748356307814841385599531367, 12.75607679016351918194283581149, 13.63442593591458264992629294860, 14.36072506639191741873194095659, 15.1445683785164815131506653586, 15.830745377285514815589299841111, 16.26106518584098265801766471248, 17.11529043542618365048803749553, 17.5551355266571120884265320388, 18.06751125547219067870948147875

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