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Properties

Label 1-1400-1400.221-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.876 - 0.481i$
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  + (−0.669 + 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

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05040754165 - 0.1965460726i\)
\(L(\frac12)\) \(\approx\) \(0.05040754165 - 0.1965460726i\)
\(L(1)\) \(\approx\) \(0.6897701654 + 0.03037274830i\)
\(L(1)\) \(\approx\) \(0.6897701654 + 0.03037274830i\)

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

−21.16528489646442054825502605837, −20.237752047758747523008581756923, −19.57066843887254634752628525635, −18.626354664230494297842447490949, −18.19464168231141213249262582370, −17.37455809522634035319760342351, −16.71575508051326044283510613388, −15.96222223900571143508776992980, −15.01496011866474639352639637329, −14.19928881149091038098358713555, −13.221178351614878345843359851849, −12.77926122733123344150535341926, −11.92201474374094767013259625714, −11.12254786321748657436018460016, −10.58092382772005451242076749574, −9.398692817180962903157451380968, −8.63786411191492077227350581533, −7.60357543821777337306254198531, −6.87323532806700930039710505118, −6.29142890525153250411250816215, −5.25389944784953375029694580484, −4.49978954595469654099244250795, −3.42270636700796521225847313099, −1.97309881959561192453743638808, −1.57976701006601387040985180496, 0.08609919867064057362923567719, 1.30718588085404076107177726118, 2.827014969446849314865472688764, 3.62024883514286036731491545547, 4.53337027246077518761368959776, 5.324803756443856987680567638014, 6.258396913724203590433194538916, 6.73128513124681640049572369998, 8.22076441277055102265249022079, 8.80077882126205906540974921502, 9.63948897153181732509635413644, 10.71876778955823224605089191098, 11.04548163180236729242715912397, 11.75549165644220289802699667427, 12.97393877139385675491624039733, 13.39707359585237373972335264217, 14.60948854560967354265368039070, 15.3591425067441801785186593811, 15.843341725567238519325848454769, 16.851018434239768592593045171088, 17.194851270046302867071005833692, 18.22242532200641199851338170818, 18.79960386407205241321985378713, 19.940218566120140137861053277750, 20.48842595440636971016074125416

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