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Properties

Label 1-1400-1400.691-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: $\chi_{1400} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267385113 + 0.1057318460i\)
\(L(\frac12)\) \(\approx\) \(1.267385113 + 0.1057318460i\)
\(L(1)\) \(\approx\) \(0.9743161410 + 0.1981787188i\)
\(L(1)\) \(\approx\) \(0.9743161410 + 0.1981787188i\)

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

−20.643727002234732811159614515474, −20.015179561269675258172352254597, −19.1358643282137264325106119527, −18.44413737043666381232140154686, −18.02086612864562900490918236433, −17.11934687261994097889969556251, −16.22836267162426684987336787240, −15.55736740875505577708310755203, −14.38103723534246001476353325902, −13.92906840408577613665062845812, −13.12974571143145402846352823040, −12.364314862983678440334344739463, −11.73360875932006043192423458737, −10.82263008192655992191997721194, −9.97397021215246996905136817225, −8.83152279054879221733903942858, −8.28687245117688572777246807613, −7.381699728947932567031074106979, −6.65112420226025664676101048667, −5.91206244202346605260058586630, −4.88843265828145345332860509593, −3.88476618580367764463635545612, −2.626857805094685289137469565871, −2.08195229406082121453703128107, −0.88577938980224363829674669780, 0.61035938735083028663648106362, 2.38467431719891076364244680208, 2.9753810344640893903255550677, 4.05561216437733379164295635876, 4.80096197543332460601118482848, 5.6518708753726929818564772840, 6.40071984110274021684867174021, 7.83373262372590462846117989282, 8.2556570451928278265984230765, 9.292998280810610384804040448101, 9.96468310267269118386086125662, 10.853966254642069696476918127333, 11.21846686081313623207015442488, 12.397295315345692878036651806788, 13.37856711664359802718641900004, 13.855196922078775515154796931111, 15.03453727611195220774718294813, 15.67521691096316232027911439954, 15.859401045179690505971741385828, 17.15006188590938379411061708580, 17.62310904286718230794758303463, 18.47881980505799035448670385654, 19.61195426676069792432880637719, 20.08457824193127162939253368288, 20.85352589451103650235623350518

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