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Properties

Label 1-1840-1840.1477-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.968 + 0.247i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.142 + 0.989i)3-s + (−0.755 − 0.654i)7-s + (−0.959 + 0.281i)9-s + (−0.540 − 0.841i)11-s + (0.654 + 0.755i)13-s + (−0.909 − 0.415i)17-s + (0.909 − 0.415i)19-s + (0.540 − 0.841i)21-s + (−0.415 − 0.909i)27-s + (0.909 + 0.415i)29-s + (−0.142 + 0.989i)31-s + (0.755 − 0.654i)33-s + (−0.959 + 0.281i)37-s + (−0.654 + 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)3-s + (−0.755 − 0.654i)7-s + (−0.959 + 0.281i)9-s + (−0.540 − 0.841i)11-s + (0.654 + 0.755i)13-s + (−0.909 − 0.415i)17-s + (0.909 − 0.415i)19-s + (0.540 − 0.841i)21-s + (−0.415 − 0.909i)27-s + (0.909 + 0.415i)29-s + (−0.142 + 0.989i)31-s + (0.755 − 0.654i)33-s + (−0.959 + 0.281i)37-s + (−0.654 + 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.968 + 0.247i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.968 + 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07069132370 + 0.5623314327i\)
\(L(\frac12)\) \(\approx\) \(0.07069132370 + 0.5623314327i\)
\(L(1)\) \(\approx\) \(0.7888922709 + 0.2607006572i\)
\(L(1)\) \(\approx\) \(0.7888922709 + 0.2607006572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.755 - 0.654i)T \)
11 \( 1 + (-0.540 - 0.841i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (0.909 - 0.415i)T \)
29 \( 1 + (0.909 + 0.415i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (-0.755 + 0.654i)T \)
61 \( 1 + (-0.989 - 0.142i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.281 + 0.959i)T \)
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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

−19.86102815114666916197980187442, −18.93385277201666251120751786108, −18.38736388494746141967755424645, −17.77599453986072392552342926144, −17.11418553607642549026884137811, −15.89925184121556701499568723205, −15.48806091627538893354467543907, −14.65340530186696463569932065643, −13.553707183470498716671462338510, −13.20462243599994690432750985303, −12.35511298849417478978883435582, −11.934808278086158006971391219284, −10.829725896303743202766659553007, −10.02818915117852250492114019378, −9.06940497296553291830411743055, −8.42729819334356740007765239559, −7.55749024867319146254060265409, −6.890782549337345913498219817441, −5.93338655682711215292504632222, −5.51627553182886463290818818916, −4.1669195362231906796386525890, −3.0879511951989382827103314728, −2.43681845765282075220486146416, −1.5292901548110423026425535701, −0.20134109030670321048294004635, 1.20232996197318244755248896004, 2.8138672636877549502946325191, 3.215332705891547598630512124784, 4.222817458182803074752130916836, 4.874655582331661555477823812847, 5.90447424735106055405296510818, 6.64480849532548912246577739454, 7.59913473198383855279970039931, 8.69348009961525442503606571776, 9.13593633951501252733812924402, 9.994658379701272966767937827631, 10.793817274089190121628974298066, 11.21233914436875509203415040559, 12.21993941964011680087509239956, 13.46840866313735221147748043716, 13.707608975233388282935049738366, 14.51010795195441523750435398278, 15.682379272551384420027653395587, 16.08888548264915892035630964719, 16.42700044117227453399940064024, 17.509301818454298835410633755488, 18.20245668622771822895678561459, 19.30093196246999509217886863280, 19.74202609335528659000322758216, 20.538079459682123644755068127833

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