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Properties

Label 1-1400-1400.339-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.986 + 0.165i$
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  + (−0.104 + 0.994i)3-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.309 − 0.951i)13-s + (0.913 + 0.406i)17-s + (0.104 + 0.994i)19-s + (0.669 + 0.743i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.913 + 0.406i)31-s + (−0.104 − 0.994i)33-s + (−0.978 − 0.207i)37-s + (0.978 − 0.207i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)3-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.309 − 0.951i)13-s + (0.913 + 0.406i)17-s + (0.104 + 0.994i)19-s + (0.669 + 0.743i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.913 + 0.406i)31-s + (−0.104 − 0.994i)33-s + (−0.978 − 0.207i)37-s + (0.978 − 0.207i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.986 + 0.165i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06032722573 + 0.7231296027i\)
\(L(\frac12)\) \(\approx\) \(0.06032722573 + 0.7231296027i\)
\(L(1)\) \(\approx\) \(0.7551493780 + 0.3615302254i\)
\(L(1)\) \(\approx\) \(0.7551493780 + 0.3615302254i\)

Euler product

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

−20.434350051979591772665887555837, −19.41399232419520577346991469903, −18.94274180695903502357548218028, −18.29506685872014606640180535854, −17.49591274674932646255406488474, −16.76419247347589795818824744109, −16.021156388925326686325785707803, −15.01837407108410538852342350963, −14.16690762260348447806093902742, −13.48777898141428218293061157246, −12.88439917228617204237983583774, −11.90154026960417207180510875450, −11.47348586914808354092783465450, −10.43348842075179333805835558814, −9.54716707094079460198588574332, −8.48177926724997858596982135879, −7.93499900914260897426706831701, −6.92516811055911783636932716818, −6.46796990374365798254967650805, −5.277127987200873346829794917710, −4.70039646234702340696077262750, −3.139384557629580002000389485782, −2.53602645917418698820222603178, −1.46141001335891750720498041055, −0.28697460734766304268748916483, 1.35112236630910954876666836323, 2.86151980951675746006743652098, 3.31969470107654114362598499742, 4.452748890212353494112629196592, 5.35894985147844819705657726660, 5.7290510513398300636169980746, 7.08147625354691051397987965011, 8.05002368900996479797389640505, 8.62829971721688440187374190473, 9.87021461684355735404679876878, 10.21782783680426936130480574456, 10.88096616158056852663954572190, 12.00208842143016298326021487415, 12.57354839727648729991074666499, 13.6512784008424968220025868068, 14.4552170348118356121345412688, 15.24893358295533762431273212610, 15.73582959869709883568760718784, 16.5693721430601140418556597184, 17.31560488062042742100061900655, 17.97952830577350528929672518282, 18.96094472701720904813768084227, 19.77345390030116504931930949613, 20.60040227279738862367676633267, 21.14903400080120244965561085869

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