Properties

Label 1-1840-1840.1389-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.715 - 0.698i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.989 + 0.142i)3-s + (−0.654 − 0.755i)7-s + (0.959 + 0.281i)9-s + (0.540 − 0.841i)11-s + (0.755 + 0.654i)13-s + (−0.415 − 0.909i)17-s + (0.909 + 0.415i)19-s + (−0.540 − 0.841i)21-s + (0.909 + 0.415i)27-s + (−0.909 + 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (0.281 − 0.959i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)3-s + (−0.654 − 0.755i)7-s + (0.959 + 0.281i)9-s + (0.540 − 0.841i)11-s + (0.755 + 0.654i)13-s + (−0.415 − 0.909i)17-s + (0.909 + 0.415i)19-s + (−0.540 − 0.841i)21-s + (0.909 + 0.415i)27-s + (−0.909 + 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (0.281 − 0.959i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.191456438 - 0.8915606856i\)
\(L(\frac12)\) \(\approx\) \(2.191456438 - 0.8915606856i\)
\(L(1)\) \(\approx\) \(1.499831494 - 0.2040770763i\)
\(L(1)\) \(\approx\) \(1.499831494 - 0.2040770763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.909 + 0.415i)T \)
29 \( 1 + (-0.909 + 0.415i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 - T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.755 + 0.654i)T \)
61 \( 1 + (0.989 - 0.142i)T \)
67 \( 1 + (-0.540 - 0.841i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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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

−20.09342304669125434781186246252, −19.633705536618369103421682616, −18.84926269578888582870591036561, −18.12282092627697920781649236733, −17.54156667498536874054921470879, −16.322570507370372584072327281695, −15.666093959138988226776630822971, −15.03137388996766005768122214603, −14.51848985736373763516560718293, −13.31081806990555715440227310368, −13.06201165252650884963454377073, −12.213431095971146208124744303646, −11.37331128577232676939703016382, −10.185573968791289078661244289767, −9.64358384025659484254282063551, −8.84706837606673133375552881841, −8.292727719787530416684574498233, −7.30770332844725274107854911729, −6.57891333882432068785070886242, −5.74384454977551291178564322, −4.62681373977270163981984740825, −3.65646865286049763705484555041, −3.04250346471555569575082571588, −2.08817201716465058338106498252, −1.23310633871469743554259181001, 0.7922384909861583856714940133, 1.82340982564789899268362284624, 2.93787262628767672510717588503, 3.70627543508551745034423215932, 4.14437285152805266971752418617, 5.41629569158424438721750777307, 6.45722895266988372580775341524, 7.16864980713924630903365492256, 7.89328219917016841777836783194, 8.925211746755261266159878068661, 9.345643263511796624270055215073, 10.11886906922202676176213375308, 11.091091102124633271582223907196, 11.72570207768985482491796103057, 13.03918775054298492775429010132, 13.410204378645101699696139283463, 14.13492416455107681499747551396, 14.654782607735260431898420952632, 15.858915776703835915558635255622, 16.23717462416444342596529431090, 16.836355995371194289258681623461, 18.15103457892657928060793689289, 18.650886424752322481721765939710, 19.5017197255836417090791767451, 19.96207711767773449375443705361

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