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Properties

Label 1-31-31.2-r0-0-0
Degree $1$
Conductor $31$
Sign $0.817 + 0.575i$
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 + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s + 6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s + 6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ 0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8457843661 + 0.2679001185i\)
\(L(\frac12)\) \(\approx\) \(0.8457843661 + 0.2679001185i\)
\(L(1)\) \(\approx\) \(1.070408848 + 0.2744754543i\)
\(L(1)\) \(\approx\) \(1.070408848 + 0.2744754543i\)

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 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
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

−37.0545848366649830621588968149, −36.03778730063649981964712062892, −33.74023848308893231034556184066, −32.723758465106805549710233990564, −31.952748045442972615343568073826, −30.539779999599055575096157222012, −29.04537212905841648562831037428, −28.45625410953668564944820627434, −26.65031954705866006226884531502, −25.96737517960782840742560102402, −23.81665881039987561569497948094, −22.230352453778010566413340556755, −21.49164921331079578596889795386, −20.39221306875086787867779537633, −19.14707703188554896128280292362, −17.41501077784915130389185468954, −15.81576715155143292637446285390, −13.98468128223777515193119356492, −13.28168763647461597663439330398, −11.07006849586849188926080336733, −10.006183367778683829854140480289, −8.97770977821202751023412989903, −5.88345510750945743765712651368, −4.19315521765601260223127809096, −2.58899933971503231217203064540, 2.759993489764024493600302113485, 5.54759538481889561407498053938, 6.63281731113214783469667843482, 8.21255669260890500382857208059, 9.67742685821920843552283041135, 12.583113048435116792796479063, 13.26391047899968339343910420375, 14.60944892709433446320653308062, 16.12229485816759716868598304741, 17.85608057244635881364273770356, 18.39151322832905350038619987433, 20.4376810526101145134056928729, 22.153528945368338138093560112108, 23.21535717927928313207569971828, 24.78082904556763323571305656246, 25.3462701510056390081342470637, 26.326382462122383330352721622559, 28.46679809010471140612483272781, 29.67108234088307875972863650524, 31.04915943791976161437395123415, 32.040030353483516231227280059073, 33.30509705867886992104321370122, 34.562068701071793296816911901679, 35.653369201196103839975872486500, 36.54664994615698913938790433681

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