Properties

Label 1-1400-1400.723-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.729 + 0.684i$
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  + (0.406 + 0.913i)3-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.951 − 0.309i)13-s + (−0.994 + 0.104i)17-s + (−0.913 − 0.406i)19-s + (0.207 + 0.978i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.104 + 0.994i)31-s + (−0.406 + 0.913i)33-s + (0.743 + 0.669i)37-s + (0.669 + 0.743i)39-s + (0.309 + 0.951i)41-s + i·43-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)3-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (0.951 − 0.309i)13-s + (−0.994 + 0.104i)17-s + (−0.913 − 0.406i)19-s + (0.207 + 0.978i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.104 + 0.994i)31-s + (−0.406 + 0.913i)33-s + (0.743 + 0.669i)37-s + (0.669 + 0.743i)39-s + (0.309 + 0.951i)41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.729 + 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5308767509 + 1.341906617i\)
\(L(\frac12)\) \(\approx\) \(0.5308767509 + 1.341906617i\)
\(L(1)\) \(\approx\) \(1.004738439 + 0.5372293191i\)
\(L(1)\) \(\approx\) \(1.004738439 + 0.5372293191i\)

Euler product

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

−20.48073822480648234562053154845, −19.64523994878818281794172081810, −18.9175553628978402776244742646, −18.47583993714728424636097495944, −17.57016194532433034081243312609, −16.81106145065782380594981689824, −16.05049370440457097446691674015, −14.93692946854415515561383203115, −14.39545857253627153582151599770, −13.50768178009182475373085283619, −13.030960526694448935022632043301, −12.13871964354445381213032789187, −11.27423620400579628198017376177, −10.71680560471599077509648984517, −9.255433220780776494633521317712, −8.77719900895765305113095650982, −8.09610322839442905722202578774, −7.032163288786521614627999380097, −6.35667301329894945376182337755, −5.75504779757352003910693355551, −4.25912752876656639779159506892, −3.578058956365619869987820269324, −2.44187970061400688658924347028, −1.66130909752426876986137542480, −0.51176796511914044327372882866, 1.46015135095183297698172802549, 2.48377076772371001796417590826, 3.5104294628640438957538474466, 4.24397234957918899136449388278, 4.95733394542417009580356094691, 6.057939799793100689410153100895, 6.860478134434654830507411253602, 8.03387054449648132364424736662, 8.70068155864585868766063326052, 9.47344822067092100080553675367, 10.120738617752071956963781506146, 11.187690021533567257066269928783, 11.47197224904981620451535354221, 12.9677124111141508177568718844, 13.35051885782042321079748542548, 14.50359844168483248041873013361, 15.00395162221267490341865258662, 15.709933377569052100474513805672, 16.39772135931685544912499862346, 17.37077175723996286544691643470, 17.82996664305494503736359733829, 19.044445102114759467266841828880, 19.79175004656064021863165769204, 20.24877156675109293003269942158, 21.17716662399409368718808311135

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