Properties

Label 1-837-837.20-r1-0-0
Degree $1$
Conductor $837$
Sign $0.928 + 0.372i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (0.374 + 0.927i)11-s + (0.961 + 0.275i)13-s + (−0.990 + 0.139i)14-s + (0.559 + 0.829i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.615 − 0.788i)20-s + (0.990 − 0.139i)22-s + (−0.990 + 0.139i)23-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (0.374 + 0.927i)11-s + (0.961 + 0.275i)13-s + (−0.990 + 0.139i)14-s + (0.559 + 0.829i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.615 − 0.788i)20-s + (0.990 − 0.139i)22-s + (−0.990 + 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.212924690 + 0.2341478684i\)
\(L(\frac12)\) \(\approx\) \(1.212924690 + 0.2341478684i\)
\(L(1)\) \(\approx\) \(0.9015274414 - 0.3275736526i\)
\(L(1)\) \(\approx\) \(0.9015274414 - 0.3275736526i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
11 \( 1 + (0.374 + 0.927i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.990 + 0.139i)T \)
29 \( 1 + (0.241 - 0.970i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.559 + 0.829i)T \)
43 \( 1 + (-0.719 - 0.694i)T \)
47 \( 1 + (-0.438 + 0.898i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.719 + 0.694i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−21.808773887741312458292938555183, −21.47664309164931567570204788068, −20.361612436243358176736514779567, −19.37239661739728348755976271045, −18.535359749477269833812586535964, −17.84167748483939175138439381521, −16.67782371238117853422076761620, −16.33380327236455590882727591859, −15.593694334881034648181388158996, −14.78059429559472701067505315478, −13.79673771984557308502054120558, −13.00334596501197889019034605425, −12.392414618879462664666768918604, −11.54332039158730838071572722581, −10.14963290046555407464506348143, −9.08617725256023295653630333005, −8.44395396561219454152090258872, −8.07024692941579832157194991825, −6.47189583517781799063588035480, −5.93285700889908328967908555690, −5.21963376328492268497112304845, −3.99531758198423783137437792444, −3.38055530063306321811366557037, −1.64251000455193880307497683981, −0.31774189137428495743018529315, 0.8773480837432390303435913632, 2.13119806141513383033965599994, 3.039684118100009324701356939249, 4.006618980775160721264973434629, 4.56532491080988756860249132603, 6.07786448265052434609205422994, 6.84260694887495765321776020620, 7.79016237432806296919477779605, 9.086264989412558947786139745719, 9.908541550051385229170291153213, 10.51247638324763146696087426415, 11.44741840265631285651019774304, 11.89670685351654039550351177299, 13.37258034729242852510595826219, 13.545172008490756731811960766341, 14.56066679228493226925093630375, 15.32825358709245671219120183444, 16.349721525449128186220377020210, 17.642999327483579579926611482248, 18.02680355389043351344323066500, 18.95205398023104866693448639713, 19.792607885946897774202539241458, 20.22981949821110319642911082136, 21.1552808613612709063649081517, 22.11938250340147380920650055248

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