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Properties

Label 1-1400-1400.1091-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.728 - 0.684i$
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  + (0.809 − 0.587i)3-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)17-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (0.809 + 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)17-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (0.809 + 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + (−0.309 + 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.154684228 - 0.8530998363i\)
\(L(\frac12)\) \(\approx\) \(2.154684228 - 0.8530998363i\)
\(L(1)\) \(\approx\) \(1.473951709 - 0.3268184837i\)
\(L(1)\) \(\approx\) \(1.473951709 - 0.3268184837i\)

Euler product

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

−20.96261610914574820068672964578, −20.10498492764084845260219969964, −19.44296526108937890178815253356, −18.82668800965418101133764666236, −17.98456788112356388622505286735, −16.903947418555528115545344583892, −16.02082008995681249424585715160, −15.89675020415423333926157959801, −14.63249157523761776112694726030, −13.94819897607011769649205821055, −13.70304802279262152532815317276, −12.463828358630448941964508079970, −11.51057561043774943667493834953, −10.87307451673735628181493450346, −9.87134488690874707595979348213, −9.15029289272927072854837041234, −8.63915854584361004498755261970, −7.6068748061725128416951306671, −6.89856124200769494326569069538, −5.64015351021761342126495217312, −4.94634644002135371126359427612, −3.75564759507136121786384435218, −3.34045688333828039625136213621, −2.23741349772281249137840508749, −1.13729522045126958683288033250, 0.9717278592721979368560034962, 1.84844271861483869240192535828, 2.87360373613983271929675883688, 3.64429287007202773652851996552, 4.60737421436070133363781369432, 5.82578809762614185681583821214, 6.55050046749185631454152685281, 7.63985719440603978856840262216, 7.987140145728804340381233581055, 8.949060908545605023819314988639, 9.8764097089112673601142869991, 10.38842696702892921238571269582, 11.78564667371131642956552832681, 12.35782472824242938971652808643, 13.04557787015490962495078003121, 13.842110865840514388803610174279, 14.73135476567235486315978974274, 15.07160449553353645484876681426, 16.102922467643720579259655675314, 17.01910728117620293813212390848, 17.99920704710581664875819648258, 18.31127257790387361785883418960, 19.33254610113119472423910751790, 19.9223078755479894209745024959, 20.662228939339545196522682377715

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