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Properties

Label 1-1400-1400.179-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.876 + 0.481i$
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  + (−0.669 + 0.743i)3-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (0.913 − 0.406i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.978 − 0.207i)31-s + (−0.669 − 0.743i)33-s + (−0.104 − 0.994i)37-s + (0.104 − 0.994i)39-s + (−0.809 + 0.587i)41-s − 43-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)3-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (0.913 − 0.406i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.978 − 0.207i)31-s + (−0.669 − 0.743i)33-s + (−0.104 − 0.994i)37-s + (0.104 − 0.994i)39-s + (−0.809 + 0.587i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.876 + 0.481i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.876 + 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.446138728 + 0.3708865672i\)
\(L(\frac12)\) \(\approx\) \(1.446138728 + 0.3708865672i\)
\(L(1)\) \(\approx\) \(0.8511540576 + 0.2158375576i\)
\(L(1)\) \(\approx\) \(0.8511540576 + 0.2158375576i\)

Euler product

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

−20.44227950594391471544870305034, −19.61573988723322484871403534543, −18.94455045604538547228848417368, −18.37703550781563911794510620659, −17.42084483539718783166513826077, −16.95126863567918562644482701969, −16.17402597222997273142800321512, −15.286250842192835492471473793613, −14.320793012643321301722488740364, −13.51503982335582458481680743991, −12.921053337564307579640482296127, −12.017887315166138803836132902739, −11.463379089409424734767345546100, −10.59360677028285547137806235725, −9.85485807137894628362779707944, −8.66797191224005693223929104462, −7.91253119373700718363381172353, −7.13032840772717121605817211621, −6.382889355649086330869955441601, −5.2621078367965859991336401085, −5.07973072453387445147417258660, −3.39442872096177753143345902546, −2.71264462696311105338789618237, −1.368103228752075395048491243488, −0.648852763818004980963165817501, 0.514057514996839438754853680538, 1.753877099838823587339292060461, 2.9625506997052580156211346577, 3.937017260747085044109908184491, 4.81943017035311963352387592644, 5.35352451996621988643836991453, 6.43214804298483865130718951824, 7.19925010973663411688587145554, 8.125952041778005546350453943639, 9.36713435439757697907790894725, 9.84070715433933227940329379968, 10.44082693385253008152722992313, 11.64057140310945046128318534645, 11.974963603317986598117661220311, 12.81582540780279683919567566707, 13.947213018623482595613192786091, 14.88598562361514383683098168914, 15.18485375592093465618286765027, 16.41953846679429016463964495661, 16.68910461705407341146258136495, 17.56717150750494633143989354599, 18.27085620715982382468784904888, 19.13463575229485219933712298701, 20.04691761830702254674577992537, 20.98765029064051220052269893826

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