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Properties

Label 1-3800-3800.739-r1-0-0
Degree $1$
Conductor $3800$
Sign $0.993 + 0.109i$
Analytic cond. $408.366$
Root an. cond. $408.366$
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  + (−0.438 + 0.898i)3-s + (−0.5 + 0.866i)7-s + (−0.615 − 0.788i)9-s + (−0.978 + 0.207i)11-s + (0.990 − 0.139i)13-s + (−0.961 + 0.275i)17-s + (−0.559 − 0.829i)21-s + (0.848 − 0.529i)23-s + (0.978 − 0.207i)27-s + (−0.961 − 0.275i)29-s + (0.104 + 0.994i)31-s + (0.241 − 0.970i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + (−0.882 + 0.469i)41-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)3-s + (−0.5 + 0.866i)7-s + (−0.615 − 0.788i)9-s + (−0.978 + 0.207i)11-s + (0.990 − 0.139i)13-s + (−0.961 + 0.275i)17-s + (−0.559 − 0.829i)21-s + (0.848 − 0.529i)23-s + (0.978 − 0.207i)27-s + (−0.961 − 0.275i)29-s + (0.104 + 0.994i)31-s + (0.241 − 0.970i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + (−0.882 + 0.469i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.993 + 0.109i$
Analytic conductor: \(408.366\)
Root analytic conductor: \(408.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3800,\ (1:\ ),\ 0.993 + 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5321480492 + 0.02925981550i\)
\(L(\frac12)\) \(\approx\) \(0.5321480492 + 0.02925981550i\)
\(L(1)\) \(\approx\) \(0.6370401738 + 0.2802539713i\)
\(L(1)\) \(\approx\) \(0.6370401738 + 0.2802539713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.438 + 0.898i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (-0.961 + 0.275i)T \)
23 \( 1 + (0.848 - 0.529i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.882 + 0.469i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.961 + 0.275i)T \)
53 \( 1 + (-0.241 - 0.970i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (-0.848 + 0.529i)T \)
67 \( 1 + (-0.559 + 0.829i)T \)
71 \( 1 + (0.997 - 0.0697i)T \)
73 \( 1 + (-0.990 - 0.139i)T \)
79 \( 1 + (-0.438 + 0.898i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.882 - 0.469i)T \)
97 \( 1 + (-0.559 - 0.829i)T \)
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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

−18.610901574029556088825149702292, −17.74366067939789518265445991227, −17.04274828244305865166689303584, −16.52074431249394762351416433678, −15.76088491606006577167960997397, −15.099252243147401892160162856647, −13.86417265768404360168817952498, −13.467329364974910359098814101282, −13.1185936925915021455720200313, −12.32216263403077894415725304619, −11.31144673359531372345116553854, −10.93592715663534790975409922227, −10.31689374723793141527465514718, −9.20233138652893882299669823071, −8.570245550827363195929929608116, −7.51827853087375555363267018533, −7.28579529821883454461340713204, −6.36138113721853361149911711753, −5.769814796105759501728305470954, −4.95149949895238810906330101201, −3.98978063453994754151559155523, −3.1417044830434852289492855799, −2.25610452180419524495404267841, −1.36167070336023006998681681495, −0.45538138190473310504393512583, 0.164204850294586800392377053071, 1.473067275252003192009550968900, 2.66481799697077442943450623384, 3.18058441606924375243578645132, 4.11963384655435993658252817120, 4.95427534493737832859654583263, 5.51017350972099340343924341496, 6.30758321738699722554731615399, 6.86150282104373137115234950675, 8.23761209724565997576920280689, 8.64660033058773216240757338508, 9.4329507116241407589906785731, 10.09858349792204391902204100004, 10.86005428191536145234107675507, 11.31645067473047658396881103914, 12.17913180081517987273292447224, 12.968506819526551321162641586732, 13.42008575721164133022489371316, 14.59694698504145359543800311537, 15.32098754339691837325032263321, 15.57116314833937926979626626620, 16.2872347017131819020121943616, 16.92341087189908176496095644821, 17.81962124114906626507406780806, 18.33785355810849634244419519990

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