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Properties

Label 1-4031-4031.273-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.996 - 0.0791i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
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  + (−0.699 + 0.715i)2-s + (−0.313 + 0.949i)3-s + (−0.0227 − 0.999i)4-s + (0.829 + 0.557i)5-s + (−0.460 − 0.887i)6-s + (0.538 + 0.842i)7-s + (0.730 + 0.682i)8-s + (−0.803 − 0.595i)9-s + (−0.979 + 0.203i)10-s + (−0.761 − 0.648i)11-s + (0.956 + 0.291i)12-s + (−0.746 + 0.665i)13-s + (−0.979 − 0.203i)14-s + (−0.789 + 0.613i)15-s + (−0.998 + 0.0455i)16-s + (−0.907 + 0.419i)17-s + ⋯
L(s)  = 1  + (−0.699 + 0.715i)2-s + (−0.313 + 0.949i)3-s + (−0.0227 − 0.999i)4-s + (0.829 + 0.557i)5-s + (−0.460 − 0.887i)6-s + (0.538 + 0.842i)7-s + (0.730 + 0.682i)8-s + (−0.803 − 0.595i)9-s + (−0.979 + 0.203i)10-s + (−0.761 − 0.648i)11-s + (0.956 + 0.291i)12-s + (−0.746 + 0.665i)13-s + (−0.979 − 0.203i)14-s + (−0.789 + 0.613i)15-s + (−0.998 + 0.0455i)16-s + (−0.907 + 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.996 - 0.0791i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.996 - 0.0791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3206614261 + 0.01271617927i\)
\(L(\frac12)\) \(\approx\) \(0.3206614261 + 0.01271617927i\)
\(L(1)\) \(\approx\) \(0.4776495252 + 0.4025214492i\)
\(L(1)\) \(\approx\) \(0.4776495252 + 0.4025214492i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.699 + 0.715i)T \)
3 \( 1 + (-0.313 + 0.949i)T \)
5 \( 1 + (0.829 + 0.557i)T \)
7 \( 1 + (0.538 + 0.842i)T \)
11 \( 1 + (-0.761 - 0.648i)T \)
13 \( 1 + (-0.746 + 0.665i)T \)
17 \( 1 + (-0.907 + 0.419i)T \)
19 \( 1 + (0.0909 - 0.995i)T \)
23 \( 1 + (-0.460 + 0.887i)T \)
31 \( 1 + (0.595 + 0.803i)T \)
37 \( 1 + (-0.789 - 0.613i)T \)
41 \( 1 + (-0.356 - 0.934i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.439 - 0.898i)T \)
53 \( 1 + (0.247 - 0.968i)T \)
59 \( 1 + (-0.854 - 0.519i)T \)
61 \( 1 + (0.356 - 0.934i)T \)
67 \( 1 + (-0.983 + 0.181i)T \)
71 \( 1 + (-0.898 + 0.439i)T \)
73 \( 1 + (0.225 - 0.974i)T \)
79 \( 1 + (-0.631 - 0.775i)T \)
83 \( 1 + (0.983 + 0.181i)T \)
89 \( 1 + (0.926 + 0.377i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.20822332846738857848556698784, −17.96903592129832777598740459365, −17.236508609469563496193895852279, −16.901394417771826451988977595947, −16.16968336087500222664276694973, −15.022441551026930028294919115085, −14.00040003894532767785566598819, −13.45679735324984520541747859905, −12.95554238982047082975466101288, −12.23384608781421289830588208269, −11.77731723128947278168222741668, −10.66361713382539691520577517387, −10.3383237839578923633763131070, −9.66424842314156825442983884699, −8.61324107258093115789863409220, −8.04812783464094381221812873428, −7.47106572133917739892881929023, −6.76493059812367238247968626314, −5.83414806946226938108468431131, −4.834185118146800549563640140426, −4.40453275356324595700350601538, −2.931789850164243097358501420897, −2.28655723127871587718512672071, −1.639965603047780151648677169833, −0.87180720102705183008746491665, 0.133611995359255459977659187859, 1.765389132598904041809377945071, 2.36169114736764261343761977055, 3.30587074275115726894542964878, 4.65940769964349158110425474811, 5.15718432404471812856437370799, 5.72626809139672006078915568542, 6.45802603005142785704317331729, 7.13229517147508808843431005401, 8.21657850257248730168497182618, 8.91651623570923404113240408193, 9.30438421660114635448974973782, 10.147552853625698856995768999961, 10.68044985648278077374863227238, 11.304760797639591529696548854040, 11.96210879117934799673912436645, 13.32651271595930884954313452763, 13.96078183654998095288114988777, 14.59353101270190034768530111775, 15.31694520423526083385361246307, 15.64124818163598394952502047804, 16.41154107536244657894003193947, 17.25467175599396693536091786572, 17.71695307702937594720641170183, 18.08643993899030170705835182708

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