Properties

Label 1-1400-1400.629-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.535 - 0.844i$
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  + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.809 + 0.587i)37-s + (−0.809 − 0.587i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)23-s + (0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.809 + 0.587i)37-s + (−0.809 − 0.587i)39-s + (0.809 − 0.587i)41-s + 43-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.952347149 - 1.073311928i\)
\(L(\frac12)\) \(\approx\) \(1.952347149 - 1.073311928i\)
\(L(1)\) \(\approx\) \(1.061050533 - 0.3390015226i\)
\(L(1)\) \(\approx\) \(1.061050533 - 0.3390015226i\)

Euler product

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

−20.929115306755887945460103699743, −20.1142806217439828585442596957, −19.21059876026107898776501676040, −18.52641958134875852018434642385, −17.533950655557402588026613221966, −16.63043320128745782566172280310, −16.46402556018242835058753008235, −15.47454345072881703865534753845, −14.55249918796163017439274706413, −14.2192462513067941085019196623, −13.033142054109225852727189753722, −12.13882797949781786506229893948, −11.318286458217387553940395647093, −10.75947828805078395747255621584, −9.94246509493576664010441872381, −8.9450929818184394540214594928, −8.624603039404778503585587280150, −7.34667725355032576358210958167, −6.154943704622629432815459852479, −5.83321647094291763049860475252, −4.65465571314489837624926493340, −3.78774017489333712253903534213, −3.32175082162978366225756456814, −1.80727784560861708935179370348, −0.7231622801997040356808576971, 0.695737256655293712972459440083, 1.35531216109057041260759396722, 2.51674172804563158190065687158, 3.380367376750758703136122567698, 4.643463826816439554399479691816, 5.50074593978163794974947585570, 6.32696146027637380491761654368, 7.18577885528418384110586558882, 7.67552296630533266225568804062, 8.84705533256251923859527509581, 9.40987389619272907706365139404, 10.69660314190389442402614349059, 11.31646561680312561666067969272, 12.111499001847088224604662871436, 12.749005328449855760703996836, 13.6797188059426338168717661711, 14.083185932445922359929006519616, 15.249142629956503100141187658441, 15.90980388666226697446058417448, 16.997426599128607714354041463957, 17.51518363139855900783596809916, 18.17031466579757621319558105986, 18.95104251244879571114460736257, 19.67001714194033850284181069436, 20.340326812882070473942973130640

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