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Properties

Label 1-1400-1400.909-r1-0-0
Degree $1$
Conductor $1400$
Sign $-0.728 + 0.684i$
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  + (0.809 − 0.587i)3-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (0.309 + 0.951i)39-s + (−0.309 + 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (0.309 + 0.951i)39-s + (−0.309 + 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1770707265 - 0.4472296039i\)
\(L(\frac12)\) \(\approx\) \(-0.1770707265 - 0.4472296039i\)
\(L(1)\) \(\approx\) \(1.047682139 - 0.3783019998i\)
\(L(1)\) \(\approx\) \(1.047682139 - 0.3783019998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−20.86029027479503585562700214828, −20.44398406594448428410130429363, −19.59148619446083619239846673154, −19.125444286664577262507792559182, −17.911621859701746743668535653445, −17.41243924716309984217232204665, −16.415708177631079619059287042380, −15.426902460213803754498996538930, −15.20300132204480092004210174432, −14.37676879947540344365019679116, −13.41052889700823493484653241945, −12.84444927589121215164479696250, −11.94141465749239034980975832658, −10.727253041787355244613997617727, −10.21371471668725783535629783703, −9.536034024788391280929008771984, −8.51778802639114697869067836688, −7.962747911567021519788849067671, −7.10993591237791329973315977131, −5.99655746123166360013902131638, −4.93442837365641865190970618475, −4.28982652290158845646401459014, −3.33722789817818519755904154542, −2.42670488634631671479185927614, −1.61864301637227190989827772225, 0.07785912791512090708182214160, 1.11290911329823561146002082443, 2.378899703959993353539302008367, 2.78255113352599829573386718132, 4.06015776471499967123385563932, 4.75328260739535044102732944782, 6.2373492851264261433687448084, 6.64237847710100716605649024652, 7.64832777610815082027645917265, 8.50345349446617012825451160389, 8.99324098998843627935282914586, 9.90704954438001856525538145110, 10.96836729024262844203798638725, 11.71878429712425327320668405478, 12.62443781595451973739677939733, 13.33746708361231356960891171296, 14.05881004491009218513009574156, 14.55656681517936796993155933963, 15.6375046200256347939905165698, 16.1856543497272688250594759455, 17.266465632346731405076716733774, 18.002323395900509285068565807568, 18.79626242957559842042832109293, 19.37901228949394437147535346690, 19.94074068323582280310844718916

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