Properties

Label 1-837-837.821-r0-0-0
Degree $1$
Conductor $837$
Sign $0.927 + 0.374i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.669 + 0.743i)10-s + (−0.997 + 0.0697i)11-s + (−0.990 + 0.139i)13-s + (−0.438 − 0.898i)14-s + (0.848 + 0.529i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.997 + 0.0697i)22-s + (0.559 − 0.829i)23-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.669 + 0.743i)10-s + (−0.997 + 0.0697i)11-s + (−0.990 + 0.139i)13-s + (−0.438 − 0.898i)14-s + (0.848 + 0.529i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.997 + 0.0697i)22-s + (0.559 − 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5955264174 + 0.1156490425i\)
\(L(\frac12)\) \(\approx\) \(0.5955264174 + 0.1156490425i\)
\(L(1)\) \(\approx\) \(0.5794695383 + 0.01662917053i\)
\(L(1)\) \(\approx\) \(0.5794695383 + 0.01662917053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (-0.997 + 0.0697i)T \)
13 \( 1 + (-0.990 + 0.139i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.559 - 0.829i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (0.374 + 0.927i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.374 - 0.927i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (0.990 + 0.139i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−21.932651893599442731732252590228, −21.127150281248292520462241762320, −20.28970192982415899387458567100, −19.37824868655037151257657726790, −19.1446355292446373024278276869, −17.96158331169111212500087270677, −17.416599939201722866611131349445, −16.65877262989882901212733819439, −15.60526724138445073592104851623, −15.06478522361453187109128093202, −14.34631114408041909994255408239, −13.08401881619050922199323748129, −12.04674140790644628381613487447, −11.14605719919652787447039288559, −10.53735754356892302683227950220, −9.99681032552483432202784325841, −8.58070712636325344329169323190, −7.93190090237859530375858170085, −7.29114326673972048668591842807, −6.533927976103061178992572022444, −5.2505313275518141500826512765, −4.12844160129071774388135719425, −2.94638547095201017616453607957, −2.01520449479968300480860932308, −0.54031605819496359477818566525, 0.79229516958253571465989336600, 2.24617125072535287145803533323, 2.84642537215812644322162927213, 4.46863307511189702214158722386, 5.19296789607513893089595060265, 6.452303948566100072513235303448, 7.60220511065968700896988287943, 8.068825076604831991874222703786, 8.93113818386203745498348561275, 9.6341107601727100750169261970, 10.82012092557164889965564006885, 11.40558050788703690447592421797, 12.48105368103488601240441744194, 12.64925853647453652236590638385, 14.42051508340808329958128137595, 15.19224823960316379930272266858, 15.93803955734903467998927391282, 16.532391987808051991094590821607, 17.543795171220869492515255631430, 18.19602874664897846042410635033, 19.07639288718558398152808854719, 19.57280021658664706444797720914, 20.6870523734970377973671294039, 20.99460000855448035534107987626, 21.9408096088704755952556432476

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