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Properties

Label 1-31-31.9-r0-0-0
Degree $1$
Conductor $31$
Sign $0.999 + 0.00222i$
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  + (−0.809 − 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (−0.104 + 0.994i)12-s + (−0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (−0.104 + 0.994i)12-s + (−0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.999 + 0.00222i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ 0.999 + 0.00222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6554674089 + 0.0007280021081i\)
\(L(\frac12)\) \(\approx\) \(0.6554674089 + 0.0007280021081i\)
\(L(1)\) \(\approx\) \(0.8287949537 + 0.02697815852i\)
\(L(1)\) \(\approx\) \(0.8287949537 + 0.02697815852i\)

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 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.104 + 0.994i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
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

−36.62180951710503200400478120794, −35.786189018350751291183895231476, −34.67655966463691943904815805798, −33.26939857420932421714620449708, −31.76763754141801505547696795324, −31.07952168970990083661404975181, −29.06406048176813209844528384434, −27.97702394928114803206234035563, −26.70828708802156378297478535269, −25.58163869110194213736425930101, −24.28702817211852782547577275791, −23.85098500147151386574802055835, −21.11981847472883910359385090190, −19.96636075835652272066513364295, −18.8394497802142786248619011841, −17.66329959862695496636846770262, −15.87892828465175943007024104820, −14.99499092204154228890446384519, −13.323424700965857289848156755314, −11.51996630368864209215613312180, −9.266506068824407895498415392169, −8.4473590942751644470298435269, −7.19028270903748450743965441993, −4.95271728153537707525623277335, −1.98829132601605799494657696959, 2.53375549830066134190223605277, 3.9955241988163664897829608375, 7.458957902275047094233536329113, 8.30135624286073415779836885454, 10.25055912822505416674286265980, 10.95326656741069251161491819798, 13.08182979585236854245239050099, 14.72799377082321837998853508372, 16.04162315963009589522886130447, 17.846347702779190438834596238698, 19.02714900809254327906793264300, 20.20914145445735977555706398718, 21.107338053179973782265572861583, 22.64515749542138313094737326260, 24.63841745043031974679881870131, 26.22427527340578477573072511272, 26.73063218536350799887946677506, 27.75300249129972543283861817814, 29.61031998397484575705064615932, 30.59516150190442147597336986489, 31.460538093618163556896661032687, 33.34554613893120470068398033851, 34.50500890708873703500932080590, 35.93855148924096073400996994877, 37.11299394054410036693969436955

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