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Properties

Label 1-1400-1400.123-r0-0-0
Degree $1$
Conductor $1400$
Sign $0.637 - 0.770i$
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  + (−0.994 − 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.951 − 0.309i)13-s + (0.406 − 0.913i)17-s + (0.104 + 0.994i)19-s + (0.743 − 0.669i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.913 − 0.406i)31-s + (0.994 − 0.104i)33-s + (0.207 − 0.978i)37-s + (−0.978 + 0.207i)39-s + (0.309 + 0.951i)41-s + i·43-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.951 − 0.309i)13-s + (0.406 − 0.913i)17-s + (0.104 + 0.994i)19-s + (0.743 − 0.669i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.913 − 0.406i)31-s + (0.994 − 0.104i)33-s + (0.207 − 0.978i)37-s + (−0.978 + 0.207i)39-s + (0.309 + 0.951i)41-s + i·43-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8688304036 - 0.4091288961i\)
\(L(\frac12)\) \(\approx\) \(0.8688304036 - 0.4091288961i\)
\(L(1)\) \(\approx\) \(0.7865597706 - 0.08999288777i\)
\(L(1)\) \(\approx\) \(0.7865597706 - 0.08999288777i\)

Euler product

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

−21.11900551220586029818093620405, −20.27929951462456633408264387761, −19.18249205098699874889449939313, −18.49712633848336061065104242665, −17.954350206250115759749333936, −17.00916235122328842836741417168, −16.51958147885886992554592430942, −15.527950743638443411750344654108, −15.22362570675276501929285700505, −13.85029363222832270770622611735, −13.124169730142141849725971999876, −12.53195729869886214454079606082, −11.49355814786936547116198602552, −10.8984747618257478311948816347, −10.35282315497886771103151808794, −9.286248780448792981841341486865, −8.4685225202557689637911207039, −7.35873742003884016232619835161, −6.72810418561838010530675603111, −5.5958537823931193346487187702, −5.30208685747692144961622137788, −4.10621066319775450977076775532, −3.30821644427887608086016930628, −1.93611147140393208291099286672, −0.904676471618706422923742590879, 0.55845357510760490044098409824, 1.65450162129496130813533673618, 2.83873338847822236324892074137, 3.95093509617537640920235788090, 4.893867356434385504067675083413, 5.675951271224022675511908010387, 6.263259293571006774529617726601, 7.452282007084717395512408013, 7.86303555526611455741549913894, 9.16891956753136120288940442356, 9.98348649865237642324798360643, 10.84344312033522614537203510589, 11.270753502651019391002396546559, 12.34240937771070072805922270718, 12.89551115875211022514760319897, 13.62116276046268448006104461170, 14.7101150549154671550803024784, 15.564690880920528399231721477022, 16.316517429819879314526696598051, 16.74224547058530321908869369691, 17.8350447808493490036599980715, 18.451986576089699645714353302189, 18.740068842720749720958775748054, 20.05640364659795637859992451327, 20.95215296768387129880809465612

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