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Properties

Label 1-1400-1400.317-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.481 - 0.876i$
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  + (−0.743 + 0.669i)3-s + (0.104 − 0.994i)9-s + (0.104 + 0.994i)11-s + (0.587 − 0.809i)13-s + (0.207 − 0.978i)17-s + (0.669 − 0.743i)19-s + (0.406 − 0.913i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.743 − 0.669i)33-s + (0.994 + 0.104i)37-s + (0.104 + 0.994i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)3-s + (0.104 − 0.994i)9-s + (0.104 + 0.994i)11-s + (0.587 − 0.809i)13-s + (0.207 − 0.978i)17-s + (0.669 − 0.743i)19-s + (0.406 − 0.913i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (−0.743 − 0.669i)33-s + (0.994 + 0.104i)37-s + (0.104 + 0.994i)39-s + (−0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4194230444 - 0.7087661005i\)
\(L(\frac12)\) \(\approx\) \(0.4194230444 - 0.7087661005i\)
\(L(1)\) \(\approx\) \(0.8210709869 + 0.01181791477i\)
\(L(1)\) \(\approx\) \(0.8210709869 + 0.01181791477i\)

Euler product

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

−21.0397705649212669575733681169, −19.88749513106076245379380282120, −19.19944613947400211600367046510, −18.56133148808650974039219133365, −17.971140381879014655560350715449, −16.961363003695399610915309416559, −16.48883111818002828606305386590, −15.82763425408485678723696012819, −14.59722973510677503831272968186, −13.88594220153296496548372134158, −13.16857629587412625362874857435, −12.41887455631263946936394572081, −11.52833857739453872443774052952, −11.069340949651687423227345278119, −10.20878540007659913977404668624, −9.09349962236852999509905428186, −8.27016356037180677344621746755, −7.46236531011627167411227947168, −6.55115662941162177218633790975, −5.87772800308494642183548805437, −5.196329814548647230442754308994, −3.97777729887266151296601473534, −3.12114529414468545510351107556, −1.67308185443141970352136479638, −1.17382887706600537822280926900, 0.20364608428059202376914419249, 1.09868972160410312500568691098, 2.555101551399535120584793126430, 3.49351516478077765838109258841, 4.53841186735710709915411995468, 5.09437497164193215702496411456, 6.01265203867908013620632794454, 6.87639208653873166037020428900, 7.697249856586317907282376891619, 8.87535080538299753774154199996, 9.63209016453272103547819970392, 10.25354510354008594035904634671, 11.1459807717666548184432163175, 11.75492171437380622267641747594, 12.610954870865748625969360707410, 13.3557690672864528293297375064, 14.47070660079359032088095378048, 15.21543418360132269526232808884, 15.79701914634232681554197507451, 16.560175453939389314697867860083, 17.33415749401511845717567291207, 18.0722886951427857409569225064, 18.51425564410257424342242964397, 19.908328316510959153474509060458, 20.45188548425663035633650958984

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