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Properties

Label 1-1400-1400.109-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  + (0.913 + 0.406i)3-s + (0.669 + 0.743i)9-s + (−0.669 + 0.743i)11-s + (0.309 − 0.951i)13-s + (0.104 − 0.994i)17-s + (−0.913 + 0.406i)19-s + (0.978 + 0.207i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.104 + 0.994i)31-s + (−0.913 + 0.406i)33-s + (0.669 + 0.743i)37-s + (0.669 − 0.743i)39-s + (0.309 − 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (0.669 + 0.743i)9-s + (−0.669 + 0.743i)11-s + (0.309 − 0.951i)13-s + (0.104 − 0.994i)17-s + (−0.913 + 0.406i)19-s + (0.978 + 0.207i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.104 + 0.994i)31-s + (−0.913 + 0.406i)33-s + (0.669 + 0.743i)37-s + (0.669 − 0.743i)39-s + (0.309 − 0.951i)41-s + 43-s + ⋯

Functional equation

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

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: $\chi_{1400} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.047166557 + 0.8098873026i\)
\(L(\frac12)\) \(\approx\) \(2.047166557 + 0.8098873026i\)
\(L(1)\) \(\approx\) \(1.438999278 + 0.2776146953i\)
\(L(1)\) \(\approx\) \(1.438999278 + 0.2776146953i\)

Euler product

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

−20.91752529965093523797201492990, −19.68729482858814373501437847777, −19.30074334944754958335480930966, −18.60435862709718476535489160001, −17.89919610206199930569436877639, −16.8514632231707337736874107832, −16.159598296027960920334940892953, −15.17400496477648220509660229093, −14.65654713773015475820215741431, −13.78386931653404457869576392663, −13.10146704705689060504623130812, −12.57387577711726727273985468600, −11.40602613922233787920318062000, −10.69430682115463137649691878497, −9.72417489776183503705174971917, −8.73318260093066852280874218838, −8.43340157135574388980714236745, −7.40569221555757238832250207094, −6.603466603007119993471767567994, −5.8142264963349866159718912259, −4.50583176864543031761522890351, −3.763432493678630191519235866466, −2.76088188861154830938965134935, −2.01863840760987949123611907806, −0.87179308582125338373728210702, 1.116189093690312747862705643573, 2.453005685561088990682597100707, 2.91517122981002222995272348038, 4.04963638243305336927236727770, 4.82486395686240616718111870470, 5.66496512607093825971481124691, 6.98327948285951148082632906194, 7.64003495809426372179746783731, 8.45387517444091303349852746904, 9.156734540139663695740045162010, 10.18644413477242443736804885790, 10.4800990215366560991989003024, 11.638419020569510911213952335344, 12.813136549880060772122267446120, 13.12573342440672001449851081756, 14.18767545963131241258307764939, 14.77915038042243250060620144706, 15.682326343530307330817665153373, 15.946053872611670525103255296331, 17.17741472242969985920111621511, 17.905553404926616093965494400014, 18.78076612792186207032351908616, 19.41025432027176130112686879535, 20.3299148651943655947370242228, 20.78857385732503460482224723172

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