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Properties

Label 1-1400-1400.1133-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.844 - 0.535i$
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  + (0.951 + 0.309i)3-s + (0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.587 − 0.809i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.951 − 0.309i)33-s + (0.587 − 0.809i)37-s + (0.809 − 0.587i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + (0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.587 − 0.809i)23-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.951 − 0.309i)33-s + (0.587 − 0.809i)37-s + (0.809 − 0.587i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.222502069 - 0.6456965401i\)
\(L(\frac12)\) \(\approx\) \(2.222502069 - 0.6456965401i\)
\(L(1)\) \(\approx\) \(1.528740982 - 0.09432361488i\)
\(L(1)\) \(\approx\) \(1.528740982 - 0.09432361488i\)

Euler product

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

−20.73956647495079767513279784556, −19.99195614076924689792581274709, −19.53907427145081375888297137695, −18.6172571756704774504670645398, −18.03501796919812209851100481779, −17.1809596450546984065901371320, −16.135505759076470092433904128306, −15.55662008005587303699212921462, −14.50692609385745560524230780622, −14.175023985561965616310996668665, −13.320037434663858724363012477899, −12.49296709014979525657575608638, −11.80091597861801724563438710671, −10.81592280562897368438298866675, −9.76761474269293017275653898139, −9.12972550839343454425730168681, −8.49398441532964626110943885429, −7.5264755930334606392272549483, −6.779041786296953465981726072891, −6.08874910276953671418756020716, −4.62194495648115892098302858955, −3.96830934837239652216209235278, −3.10519880967975259679749874649, −1.91542711850343238904415094028, −1.39690868635516003192577156473, 0.81202319746836020029072506614, 2.14279948434151013393703751677, 2.84227550964373489320021835400, 3.97761573817512981623930990243, 4.366978305425638948997730512689, 5.74579783426203865952288885794, 6.52648068904775277825051223809, 7.560024557444208043048345919528, 8.40928750454449621767841361891, 8.93476330020314278926305177606, 9.73366631070881888416091093331, 10.72558038637727434720717939779, 11.262616578408220306230629856135, 12.49142788179443084987113856670, 13.27820139120111190458404800593, 13.82553959712122374111231822546, 14.70346992081168141309768322485, 15.37004964520249615415221316623, 15.99453125859827992401123533792, 16.892953949417481463461278481895, 17.772704391902098450379708326011, 18.57929568585646554391179285104, 19.463461573788319125821580657213, 19.90909357997860804386967831077, 20.64591327390633604713795785502

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