Properties

Label 1-31-31.18-r0-0-0
Degree $1$
Conductor $31$
Sign $0.155 - 0.987i$
Analytic cond. $0.143963$
Root an. cond. $0.143963$
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  + (0.309 − 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)10-s + (0.913 − 0.406i)11-s + (−0.978 + 0.207i)12-s + (−0.978 − 0.207i)13-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)10-s + (0.913 − 0.406i)11-s + (−0.978 + 0.207i)12-s + (−0.978 − 0.207i)13-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ 0.155 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6756657285 - 0.5776039891i\)
\(L(\frac12)\) \(\approx\) \(0.6756657285 - 0.5776039891i\)
\(L(1)\) \(\approx\) \(0.9466239498 - 0.5760649265i\)
\(L(1)\) \(\approx\) \(0.9466239498 - 0.5760649265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.978 + 0.207i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
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

−36.46841241280649329799470010070, −36.17193781837376906572514859064, −34.46477938487047216566875458989, −33.10321064241311408392549515490, −32.339360498734148490628773438617, −31.48696252383850320706497148183, −30.09998200434084300734741158210, −27.771825679967877523368566115733, −27.05549716005993362074279443405, −25.79875751473201848093177300463, −24.64578807537855935878148189073, −23.40542431989301985708043711682, −22.05689469312930206108918034226, −20.598321831596921559435209529356, −19.467469400239010781445423815238, −17.04427187714010050963025633500, −16.40171818278226499151970449319, −14.94212877111189000353165290445, −13.89869062915112927556684950320, −12.32516288098414629694363362447, −9.86820992216816489589052736577, −8.55667006339715712275534777112, −7.21953062969992089326348503750, −4.82776541557434221420100060105, −3.85538862000002535765345155152, 2.26173985232878919099539228391, 3.605754751216236908346997776853, 6.17401232580066548652735631454, 8.17947387216985581853866804797, 9.72774467302622911284971283302, 11.639210467058220961159366883374, 12.52189038143484739139124825745, 14.25816822969707008442666371255, 15.00667848942998652788511647041, 17.818103654721401176372607157623, 19.18371877292355200069124583341, 19.44602046661103112065469116465, 21.32950820760234141486076887864, 22.47095610068285378340165263376, 23.810159784990289364464943417007, 25.21167947532579463280691103415, 26.74830677672859430859784633550, 27.965236292829883063114274070749, 29.67980776977912572757483063903, 30.21068024239181020196484360145, 31.51291275725207496643567390610, 32.14609240236512279102157033300, 34.38146975391731306504995047522, 35.487149214720580211009000838153, 36.8788987498810675440902823508

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