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Properties

Label 1-3800-3800.1203-r0-0-0
Degree $1$
Conductor $3800$
Sign $0.00817 - 0.999i$
Analytic cond. $17.6471$
Root an. cond. $17.6471$
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  + (−0.927 − 0.374i)3-s + (0.866 − 0.5i)7-s + (0.719 + 0.694i)9-s + (0.913 − 0.406i)11-s + (0.970 + 0.241i)13-s + (−0.469 + 0.882i)17-s + (−0.990 + 0.139i)21-s + (0.829 − 0.559i)23-s + (−0.406 − 0.913i)27-s + (−0.882 + 0.469i)29-s + (0.978 − 0.207i)31-s + (−0.999 + 0.0348i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + (−0.997 − 0.0697i)41-s + ⋯
L(s)  = 1  + (−0.927 − 0.374i)3-s + (0.866 − 0.5i)7-s + (0.719 + 0.694i)9-s + (0.913 − 0.406i)11-s + (0.970 + 0.241i)13-s + (−0.469 + 0.882i)17-s + (−0.990 + 0.139i)21-s + (0.829 − 0.559i)23-s + (−0.406 − 0.913i)27-s + (−0.882 + 0.469i)29-s + (0.978 − 0.207i)31-s + (−0.999 + 0.0348i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + (−0.997 − 0.0697i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00817 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00817 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.00817 - 0.999i$
Analytic conductor: \(17.6471\)
Root analytic conductor: \(17.6471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3800,\ (0:\ ),\ 0.00817 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9919749227 - 1.000114249i\)
\(L(\frac12)\) \(\approx\) \(0.9919749227 - 1.000114249i\)
\(L(1)\) \(\approx\) \(0.9326130250 - 0.2518379889i\)
\(L(1)\) \(\approx\) \(0.9326130250 - 0.2518379889i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.927 - 0.374i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.970 + 0.241i)T \)
17 \( 1 + (-0.469 + 0.882i)T \)
23 \( 1 + (0.829 - 0.559i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.997 - 0.0697i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.469 - 0.882i)T \)
53 \( 1 + (-0.999 - 0.0348i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (-0.559 - 0.829i)T \)
67 \( 1 + (0.139 - 0.990i)T \)
71 \( 1 + (0.615 + 0.788i)T \)
73 \( 1 + (0.970 - 0.241i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 + (0.997 - 0.0697i)T \)
97 \( 1 + (-0.139 - 0.990i)T \)
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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

−18.55825420738944766081031625125, −17.93890321070880053359656267336, −17.43092410554416351943170929494, −16.828939888014318446163679398473, −16.01166028045512493090585634562, −15.29915761425435045743854202726, −14.962044674453297663999304960171, −13.928154130434033747970902272564, −13.25704040879422073445381137896, −12.337468932970885205528868367432, −11.62045274068022770231256890154, −11.35039713375571222036820711421, −10.61335394894600250152757623276, −9.65169740938954656336275173020, −9.13394588803256623949227050583, −8.32558942506445003361285476998, −7.39747862926807187849464504452, −6.577920147320229564727853235854, −6.01728851129408003095139980726, −5.07663181872559909526770906905, −4.67038080942511825126861643467, −3.80013749322594333328166506430, −2.88639845606137036303141952306, −1.59590949063726795632444650727, −1.10567213690770946255367893321, 0.5147641543518491372267653616, 1.50578925917957569986704109247, 1.89156651559136180452351585386, 3.47007843136439151460527811579, 4.1045182488520060329243577780, 4.90748621264592140489116403409, 5.612825579882300729935405144027, 6.64803021266649503968450546148, 6.72479179463464948864204339170, 7.92339223864494603738043489518, 8.48090600505162326999574675831, 9.293283103996502226189583599513, 10.43243931642150573388562399874, 10.921066745365301595162459886535, 11.38186873014237637964471402304, 12.10378213448904281773442883396, 12.8898164733174705355208898479, 13.600108841196503217301298690570, 14.160697758988052539025753217624, 15.039838332573490228938052552882, 15.75557296539194664250474611441, 16.657729808104674536862729272196, 17.12662693180445109093742263826, 17.50283742966928178423860655199, 18.54598721509166723017162629

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