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Properties

Label 1-1400-1400.229-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.996 - 0.0847i$
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  + (0.978 + 0.207i)3-s + (0.913 + 0.406i)9-s + (−0.913 + 0.406i)11-s + (0.809 − 0.587i)13-s + (0.669 + 0.743i)17-s + (−0.978 + 0.207i)19-s + (0.104 − 0.994i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.669 − 0.743i)31-s + (−0.978 + 0.207i)33-s + (0.913 + 0.406i)37-s + (0.913 − 0.406i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)3-s + (0.913 + 0.406i)9-s + (−0.913 + 0.406i)11-s + (0.809 − 0.587i)13-s + (0.669 + 0.743i)17-s + (−0.978 + 0.207i)19-s + (0.104 − 0.994i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.669 − 0.743i)31-s + (−0.978 + 0.207i)33-s + (0.913 + 0.406i)37-s + (0.913 − 0.406i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.996 - 0.0847i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.996 - 0.0847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.356193865 - 0.1424930268i\)
\(L(\frac12)\) \(\approx\) \(3.356193865 - 0.1424930268i\)
\(L(1)\) \(\approx\) \(1.559969821 + 0.06242334538i\)
\(L(1)\) \(\approx\) \(1.559969821 + 0.06242334538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−20.69100909627636991577763519914, −19.82169110481090311254356420969, −19.119255517605524585697118625116, −18.44895906118043442814145690212, −17.916015350660540010184229673981, −16.655956867177917873128288790156, −15.9552227934198623780400208529, −15.34226752619353475095540219213, −14.3153645118156359176089430194, −13.8990928015705518737721756955, −12.98193517227941312692672185584, −12.493203070557993652819717238097, −11.20951538127048000361958173636, −10.6473484955493063377306271010, −9.441426383412986834247933936848, −9.03590214638580193168146916011, −8.02368109100763941850096074291, −7.48485789466880579530152000216, −6.52046188039662564237359829418, −5.54771379054389802275189164820, −4.48992257768823497671363293497, −3.53571872086874047841642306059, −2.819556384526109203920069112745, −1.83939209853637146445641942703, −0.84236133727915768353606866495, 0.68900213337822017065530869072, 1.99797065161083412621994964992, 2.648599201944270930030434166, 3.73988201763978619530959329958, 4.34574791193392653747472600240, 5.50278804091667887513035648462, 6.36294726384627898453305628168, 7.58781951549810517480878639652, 8.06581601862747458502792351196, 8.79115322157041458299993406371, 9.77814672198389650919047086446, 10.448695425885012764539292497538, 11.08560930654114138233402599478, 12.61969891720798611789884438746, 12.84856596507825165163386682933, 13.73791445133711869081083526497, 14.66980633829869597058889039934, 15.164512012519351399524921434847, 15.887095738151396682526171999199, 16.71238656193357931615604858439, 17.64765463667780934813581741501, 18.69217110972867638226637297807, 18.90830107025677550469465585150, 19.99083104196772969975385876347, 20.72952905235234519140808958784

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