Properties

Label 1-837-837.185-r0-0-0
Degree $1$
Conductor $837$
Sign $0.835 + 0.549i$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 + 0.984i)20-s + (0.939 + 0.342i)22-s + (0.173 + 0.984i)23-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 + 0.984i)20-s + (0.939 + 0.342i)22-s + (0.173 + 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8709531660 + 0.2607462611i\)
\(L(\frac12)\) \(\approx\) \(0.8709531660 + 0.2607462611i\)
\(L(1)\) \(\approx\) \(0.7848576124 - 0.06949849208i\)
\(L(1)\) \(\approx\) \(0.7848576124 - 0.06949849208i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−21.91230680678020066825141346695, −21.34119159212544971425037725142, −20.3883171170349682971993845036, −19.48779176641212813977704465632, −18.687174982997054625206523785891, −17.875603051787052141982708611487, −17.44236500505214773986641692273, −16.56800397332976628103341919741, −15.60087295973143814695049053054, −15.063744977196404411294875270544, −14.17885712068970102321828117450, −13.14425991285974222626737788607, −12.46574251337136966627609383037, −11.10764937464325170363965594210, −10.36447976138040900947386570787, −9.6170707599353117145314155664, −8.59329308368015120649642514986, −8.305075021467062717417894879060, −6.97966259500061129813092868228, −6.05915027579688028842769885678, −5.40364527909090751744176230362, −4.671915734604687302582603779739, −2.52595314026426965044692659931, −2.1670022722022907194460385120, −0.55300282464535676565610914452, 1.20007889348555435612673045060, 2.184640348690806145273015065625, 2.98018816579734573741365539936, 4.23320078013472511607650133908, 5.17089786390699748688477349890, 6.64903284710751425861095666471, 7.25528961984812691069582349483, 8.124678064699752588796926330188, 9.31114410668615330214589454773, 9.95426023624734057839387132088, 10.545352014350528026738186381282, 11.35585598398275966017372677306, 12.3763925015385634865812689006, 13.325236575976103088266554208665, 13.84870615228615529713332874169, 14.90576927011356029213818669853, 16.14061496134696383879456854637, 16.84963889045352974598434550763, 17.58708192713454471074744737872, 18.13847649757602705463741411572, 18.980420116052299358836205405378, 19.8553546946102926792302562888, 20.66725643222390281909423400143, 21.19669460140498151686482721429, 21.96722972559163673946751791178

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