Properties

Label 1-1400-1400.69-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: $\chi_{1400} (69, \cdot )$
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

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

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