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Properties

Label 1-1400-1400.283-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.0847 + 0.996i$
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  + (0.207 + 0.978i)3-s + (−0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (0.587 − 0.809i)13-s + (0.743 + 0.669i)17-s + (−0.978 − 0.207i)19-s + (−0.994 + 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (0.669 − 0.743i)31-s + (−0.207 + 0.978i)33-s + (0.406 + 0.913i)37-s + (0.913 + 0.406i)39-s + (0.809 + 0.587i)41-s + i·43-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)3-s + (−0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (0.587 − 0.809i)13-s + (0.743 + 0.669i)17-s + (−0.978 − 0.207i)19-s + (−0.994 + 0.104i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (0.669 − 0.743i)31-s + (−0.207 + 0.978i)33-s + (0.406 + 0.913i)37-s + (0.913 + 0.406i)39-s + (0.809 + 0.587i)41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.0847 + 0.996i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.0847 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.617450559 + 1.760883590i\)
\(L(\frac12)\) \(\approx\) \(1.617450559 + 1.760883590i\)
\(L(1)\) \(\approx\) \(1.136795761 + 0.4542831505i\)
\(L(1)\) \(\approx\) \(1.136795761 + 0.4542831505i\)

Euler product

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

−20.34724385177233353574634407133, −19.3309757037383242603320607234, −19.128502170277648622787564399485, −18.16940843699825452078306248795, −17.56098089511002795508479188148, −16.60865185084619820037260484660, −16.08614436483621074667718659568, −14.79512356525818935315378750193, −14.09545988157176555851693052221, −13.78068591235442367685684954331, −12.6011071769177089919839827956, −12.0883057637635413347069758831, −11.351689887887933023076059982656, −10.44877548109481598300986041328, −9.21518194938908321785115413559, −8.72590526320634856958816538197, −7.857239818655819804415622726750, −6.92968992188060612632144352425, −6.32860391069182459980274782554, −5.546864277082482095501421998476, −4.21145438691850571632972616937, −3.41937845998042207049462872574, −2.31820739457984831093426218948, −1.45357393676621505056101249728, −0.55454205548032313336122466861, 0.86559769513967672294063307231, 2.14833843238578599460979866948, 3.16606521251465852390594449378, 4.06334113772326058996147029993, 4.57461504346775032097972144650, 5.89504620241451900404426981969, 6.25617095520604554030777606761, 7.79361948782936952408542940658, 8.31037624413363124248100159361, 9.260287162812056625530640143368, 9.99372667122537634001673920124, 10.59739481230268888607455915833, 11.50969102230321052486651411860, 12.25554507699892710128613914095, 13.289882706896888041109629034251, 14.07018178571815650233874489189, 14.98927927479780725360577289062, 15.27619617329243140804599554400, 16.26892106462981929596771728436, 17.02169640287943615188226968352, 17.53014631036610076512708250539, 18.61159642260252266792499363066, 19.580357519848233338877512989134, 20.01912487434226579399070243561, 20.9101880896443421497780092761

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