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Properties

Label 1-31-31.13-r1-0-0
Degree $1$
Conductor $31$
Sign $-0.136 + 0.990i$
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  + (0.309 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $-0.136 + 0.990i$
Analytic conductor: \(3.33141\)
Root analytic conductor: \(3.33141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ -0.136 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4091961996 + 0.4696186391i\)
\(L(\frac12)\) \(\approx\) \(0.4091961996 + 0.4696186391i\)
\(L(1)\) \(\approx\) \(0.6966425275 + 0.07869486427i\)
\(L(1)\) \(\approx\) \(0.6966425275 + 0.07869486427i\)

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

−35.83284354285789261804393482343, −35.07575009642523564690689552083, −33.77420373882072856098162870757, −32.708151044443569429677812850035, −31.369278415875449830160356137848, −30.2176090625341776688867152558, −28.69663759769649579079528395759, −27.41104644385795494101448963297, −25.972186861998964098591452784498, −24.500683457379174766655103544271, −23.608590386111905738985523241683, −22.98207651727421599619997074568, −21.0992508598377282837706286563, −19.29065816997542433220970226951, −17.714161306338011999583994178722, −16.71288607038097140352454808856, −15.649153599907900810794755135337, −13.43671539175788866473605398856, −12.90864821742007167264492121168, −11.025350578637230731261993987943, −8.50221248490851660627576607349, −7.34183466209867228956984988019, −5.80294919096785555818731175911, −4.257359605049282262465608592655, −0.4309409368755914932724689535, 2.8317355372012336738717280039, 4.508734661225623679924903634771, 6.143281994163183992511978544442, 8.93270132033764653484673888255, 10.55204718278469759899632774022, 11.36022294981805130314356473855, 12.74826154167797804768926685114, 14.79326033851665240093582874572, 15.74175145835883606323267547242, 17.98646965392527456687985487554, 18.82064082379485176718671086767, 20.60979759463193952806976180204, 21.71767358240838251310342805734, 22.6972067688452949437004356122, 23.605066477598504230825045112764, 26.02194908366379233407718661270, 27.28449491529824232823961251793, 28.27966067372378265914154296004, 29.23220865690790798542088337987, 30.83525008421448638861649241207, 31.631412582603412820452072079456, 33.14736664913955056012400172330, 34.2787807976024235094865937016, 35.56825280117900623725268251030, 37.456702369741410259816453512592

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