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Properties

Label 1-1400-1400.683-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.588 + 0.808i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8071651376 + 0.4109950018i\)
\(L(\frac12)\) \(\approx\) \(0.8071651376 + 0.4109950018i\)
\(L(1)\) \(\approx\) \(0.8707452268 - 0.09857973252i\)
\(L(1)\) \(\approx\) \(0.8707452268 - 0.09857973252i\)

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

−20.59469277420242557104545209157, −19.949112365641460949520776566717, −19.541139718802285033560415662555, −18.14745828331826286681989158869, −17.65718949025346366443479235818, −16.73842591411492558320980093716, −16.24727160063897483288831751839, −15.3278824601404559041239631840, −14.738218165351114128866622451427, −14.01572380162464621280769706721, −13.03347407617353903847986888214, −12.018605649417696101739578299133, −11.43064631094991881895379656575, −10.55196556282915567254850572984, −9.90188767029347534475021814804, −9.07036761284101442526355616647, −8.404534782990993520397427663037, −7.32551196168219575689211657433, −6.26517190412015233373296637888, −5.555505861843669977870872581596, −4.69320125508062855494965461833, −3.78813202893647197550119241902, −3.126176451334258254233853799377, −1.87857059729130690902495138268, −0.37258143768955905927793721520, 1.18022463636111871428826458256, 1.99269843244834585442801020003, 2.92369471513475027177937403121, 4.16383617141239134534334420146, 5.03029469573524293575024031651, 6.084965152902547145086075422088, 6.80064061598177444710492970303, 7.40690777431409244920969947815, 8.31655657595121299003612723035, 9.26749898863917343790268889849, 9.92653774823875801905476416811, 11.24280969443350506609959083882, 11.80034578551430494504261593673, 12.29588158315770898232904028305, 13.337030208915312485497095707433, 14.02881157549690122131723347435, 14.534332940311990061062385946976, 15.69024353904789926080725377185, 16.54096058812112312583044666664, 17.237130643902678756155376244157, 17.96335356574275182412996314749, 18.53978223416541628079765957353, 19.52613585955756601516450358560, 19.89502268250473430982954870460, 20.72810901493441786185366540139

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