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Properties

Label 1-31-31.11-r1-0-0
Degree $1$
Conductor $31$
Sign $-0.996 - 0.0846i$
Analytic cond. $3.33141$
Root an. cond. $3.33141$
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  + (−0.809 + 0.587i)2-s + (0.104 − 0.994i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (−0.669 − 0.743i)11-s + (−0.913 − 0.406i)12-s + (−0.913 + 0.406i)13-s + (0.669 − 0.743i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.669 + 0.743i)17-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.104 − 0.994i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (−0.669 − 0.743i)11-s + (−0.913 − 0.406i)12-s + (−0.913 + 0.406i)13-s + (0.669 − 0.743i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.669 + 0.743i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $-0.996 - 0.0846i$
Analytic conductor: \(3.33141\)
Root analytic conductor: \(3.33141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ -0.996 - 0.0846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003498729413 + 0.008252757005i\)
\(L(\frac12)\) \(\approx\) \(0.0003498729413 + 0.008252757005i\)
\(L(1)\) \(\approx\) \(0.4398587578 + 0.01522644710i\)
\(L(1)\) \(\approx\) \(0.4398587578 + 0.01522644710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.913 - 0.406i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−37.09155526460713331911116549345, −36.05925963428187926730908690110, −34.951731325736862690914944839970, −33.48157646957508902140324830842, −32.02922369831403211250630634277, −31.14787776465435233359724795543, −29.17338473455325748694742682607, −28.38110459910394332347236477423, −27.23963538490566698883026537025, −26.28345832697436937006135037099, −25.02457155367068248982921554159, −22.91343902109190312026084396628, −21.62957826280450134932380578482, −20.13718496157792082973260230257, −19.81464936543266176905251163372, −17.70577790456946450353508850809, −16.34067986595775149446132695347, −15.60563444160709632332367528324, −13.08077055796011579185861908289, −11.705121106528410136018402775393, −10.04434646002069926894817379707, −9.21384182015058663386856535577, −7.5754326059981839336884181328, −4.73709792159417020058611490687, −3.04426864322953854993249123545, 0.00694396042762875805059456551, 2.64624139332603607870172746235, 6.094978255458314721019968875304, 7.1397730688944096977085936582, 8.43561437445402297351867029571, 10.25895251167032230358467296329, 11.87115921427556049504369406669, 13.73936454689426900144135318683, 15.134624687859589039692054857158, 16.54635396481488270179249202990, 18.15356986044038416004508038490, 19.00948219167128024835487233870, 19.81083534456835475851540121104, 22.39651333194633009659633951223, 23.636672228827682544944178541610, 24.68929072378361729826700043440, 26.07315484545986642767397242799, 26.73017310345942281145818473972, 28.697011094727440199302352801998, 29.39494331044338654318843999907, 30.97705204403406646811731143765, 32.2629946941892115263703159883, 34.09322865389823448355129093414, 34.817419460849600766325438953, 35.70829349002047768880615145090

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