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Properties

Label 1-1400-1400.1059-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.637 + 0.770i$
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  + (0.104 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327430838 + 0.6250820773i\)
\(L(\frac12)\) \(\approx\) \(1.327430838 + 0.6250820773i\)
\(L(1)\) \(\approx\) \(0.9037049203 + 0.2614417726i\)
\(L(1)\) \(\approx\) \(0.9037049203 + 0.2614417726i\)

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

−20.27197421801797884728838634848, −19.77355557489732037945802578697, −18.872897087708986866951002427371, −18.3051339637781281692947446942, −17.64947419446077556169493996115, −16.88827256317748351436158140103, −15.88738379297316022030723805900, −15.21680618364989561001834909793, −14.20558625705164680189470033085, −13.46840615755236555839360852695, −13.038262061355979072223587663, −12.09926761329420088773880662146, −11.29598078967944905532426132587, −10.70938712180773263338715505896, −9.3404575082995646612597665596, −8.82610349747301467531524733086, −7.864217776699505365344907339419, −7.037758011154910971573645269503, −6.55629475954259958228363108968, −5.40706447511274941518173696397, −4.655895276568119126624753906508, −3.34394199269481318790287069156, −2.45163845714482955288436425954, −1.6830957699635613576781974976, −0.496054784280158850307303017007, 0.51778147719486035441760851383, 2.10533501038586279364459833092, 3.03126603602389999934236551247, 3.799468310899950976367246904080, 4.78674471353379946334401189838, 5.49600036842195324579786417116, 6.292145592935693163377596511018, 7.543593898236168732092244510768, 8.45817127823055636391167944753, 8.873329632567057252670326875672, 10.23838012486762875996815232547, 10.41494299862129621756262205189, 11.2165196718934907791570932237, 12.305417052654284146933880875283, 13.1156280033990529632250056761, 13.918420289685523786448053913481, 14.82723222655691864444349602553, 15.46200072165758240955621169864, 16.02735556742111768628033930419, 16.83751420833008149020011001402, 17.63736312229749004180107427320, 18.39109617457823443387172735145, 19.33941205096666338594990037783, 20.12450890974502217502003475361, 20.885929549777451228936814469428

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