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Properties

Label 1-1400-1400.381-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: $\chi_{1400} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.996 + 0.0847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4118082390 + 0.01748403245i\)
\(L(\frac12)\) \(\approx\) \(0.4118082390 + 0.01748403245i\)
\(L(1)\) \(\approx\) \(0.5549724434 + 0.01059725823i\)
\(L(1)\) \(\approx\) \(0.5549724434 + 0.01059725823i\)

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

−20.804203310153312137896240667914, −19.694241596469280263610204052441, −18.98057489879072611807043015370, −18.2672042316079687394049159330, −17.50453901307360287377459039162, −16.98730259610942799900228274758, −16.06459371025497725344252031550, −15.47756198809850754382367022419, −14.58888721542059256759808034482, −13.44163172678038280227489441264, −12.963641427244508401599155909813, −12.03752107103959945150475467111, −11.44931625941434822811886991870, −10.61124126928968880184852186365, −9.87834090637779094994384581466, −9.06559751487540161468210642929, −7.74552802366437989442690459519, −7.257730006900822144528592606489, −6.35545279157897128904864753489, −5.464968083431949652453586634530, −4.75386528997173365420064824848, −3.95985796022884106885937108247, −2.43993662120465974034945163590, −1.78081393791825782796030395960, −0.27743792194513749247405654404, 0.2909004840329622737453670813, 1.654269625356045497017000483869, 2.74731088565995323486368913690, 3.87760918662376074910601702399, 4.85940139209316035457688573814, 5.39563818382560195554507416394, 6.39348371713980629335182370000, 7.04566754883573018452528007311, 8.12512358633479142086325579666, 8.90859255570356935811187019100, 10.121229330300905249027485171261, 10.586089516678401140505542918387, 11.17963138246971232370115875716, 12.34985860034594881786750006359, 12.72825354779629309889008973753, 13.54638415225109305904513846260, 14.91347095980658198468841250404, 15.189803237907694089901558238847, 16.35431746703857291173027554442, 16.673292045684963015779379588394, 17.73946670445350079650592202437, 18.08515535697754775690876685096, 19.04631663219938711451011176612, 19.84107757250270544345295307257, 20.75889629573198159086324922058

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