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Properties

Label 1-1400-1400.1117-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.588 + 0.808i$
Analytic cond. $150.450$
Root an. cond. $150.450$
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  + (−0.207 − 0.978i)3-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (0.587 − 0.809i)13-s + (0.743 + 0.669i)17-s + (−0.978 − 0.207i)19-s + (−0.994 + 0.104i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (0.669 − 0.743i)31-s + (−0.207 + 0.978i)33-s + (−0.406 − 0.913i)37-s + (−0.913 − 0.406i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)3-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (0.587 − 0.809i)13-s + (0.743 + 0.669i)17-s + (−0.978 − 0.207i)19-s + (−0.994 + 0.104i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (0.669 − 0.743i)31-s + (−0.207 + 0.978i)33-s + (−0.406 − 0.913i)37-s + (−0.913 − 0.406i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3939339271 + 0.2005845738i\)
\(L(\frac12)\) \(\approx\) \(0.3939339271 + 0.2005845738i\)
\(L(1)\) \(\approx\) \(0.7401277727 - 0.2883878553i\)
\(L(1)\) \(\approx\) \(0.7401277727 - 0.2883878553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.207 - 0.978i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.743 + 0.669i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.406 - 0.913i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−20.54825249667481267583375854962, −20.0427156294561032085094823715, −18.83614557858354060579519547528, −18.29873560252993310850755702825, −17.31485168993324141752778217711, −16.62291281189874287959770306980, −15.88831476881629167327780016237, −15.41333255605383742432112315280, −14.36772834816652099781687011775, −13.88703458833837684638951732783, −12.72573550389085562599389564981, −11.92260068856661396724453602999, −11.19049990963117631929260696496, −10.204043234880692568644715791104, −9.954950100443213028245513056536, −8.74192493929475581803001623464, −8.252997024676492021673841802937, −7.00268935190522042937388840530, −6.16340921033396220389842902200, −5.1944086365209790872984237092, −4.58122441177166977362789099777, −3.6360059209654801807559256283, −2.78245677136968753440433202042, −1.60712512717027367327960686578, −0.11241413896585938436421437027, 0.74676635809943598795186743940, 1.87242927109272307777989407137, 2.730984409175532584790340464198, 3.70601616144772560524976862732, 4.95698992968055159932467864400, 5.98946113777202545970971049585, 6.225318877365856465429388848714, 7.62248919042085939782038545213, 8.02388967881225033205437734531, 8.7336778795287439101169578770, 10.11513253215960326867242056257, 10.69304561600083538614545094089, 11.57929460024175872102385965844, 12.39099023233801180189989334023, 13.12193536557591202509795760527, 13.591823001363498482609092360750, 14.56234276150007090767438139983, 15.41586248624024436850579024302, 16.254227770943456269370396850239, 17.10350673089472885990498615382, 17.82645065400159590925700298090, 18.39521420504842682949083998419, 19.25038600763214468249308436489, 19.671038496693897909126661372675, 20.86332943934219157210342409079

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