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Properties

Label 1-4031-4031.2116-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.950 + 0.310i$
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  + (0.829 − 0.557i)2-s + (0.648 + 0.761i)3-s + (0.377 − 0.926i)4-s + (0.803 − 0.595i)5-s + (0.962 + 0.269i)6-s + (−0.247 + 0.968i)7-s + (−0.203 − 0.979i)8-s + (−0.158 + 0.987i)9-s + (0.334 − 0.942i)10-s + (−0.538 + 0.842i)11-s + (0.949 − 0.313i)12-s + (0.983 − 0.181i)13-s + (0.334 + 0.942i)14-s + (0.974 + 0.225i)15-s + (−0.715 − 0.699i)16-s + (0.877 − 0.480i)17-s + ⋯
L(s)  = 1  + (0.829 − 0.557i)2-s + (0.648 + 0.761i)3-s + (0.377 − 0.926i)4-s + (0.803 − 0.595i)5-s + (0.962 + 0.269i)6-s + (−0.247 + 0.968i)7-s + (−0.203 − 0.979i)8-s + (−0.158 + 0.987i)9-s + (0.334 − 0.942i)10-s + (−0.538 + 0.842i)11-s + (0.949 − 0.313i)12-s + (0.983 − 0.181i)13-s + (0.334 + 0.942i)14-s + (0.974 + 0.225i)15-s + (−0.715 − 0.699i)16-s + (0.877 − 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (2116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.950 + 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.472328329 + 0.7115419803i\)
\(L(\frac12)\) \(\approx\) \(4.472328329 + 0.7115419803i\)
\(L(1)\) \(\approx\) \(2.392179336 - 0.04995436399i\)
\(L(1)\) \(\approx\) \(2.392179336 - 0.04995436399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.829 - 0.557i)T \)
3 \( 1 + (0.648 + 0.761i)T \)
5 \( 1 + (0.803 - 0.595i)T \)
7 \( 1 + (-0.247 + 0.968i)T \)
11 \( 1 + (-0.538 + 0.842i)T \)
13 \( 1 + (0.983 - 0.181i)T \)
17 \( 1 + (0.877 - 0.480i)T \)
19 \( 1 + (-0.0227 + 0.999i)T \)
23 \( 1 + (0.962 - 0.269i)T \)
31 \( 1 + (0.158 + 0.987i)T \)
37 \( 1 + (0.974 - 0.225i)T \)
41 \( 1 + (-0.995 + 0.0909i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.113 + 0.993i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (-0.990 - 0.136i)T \)
61 \( 1 + (-0.995 - 0.0909i)T \)
67 \( 1 + (-0.998 + 0.0455i)T \)
71 \( 1 + (0.113 + 0.993i)T \)
73 \( 1 + (-0.746 + 0.665i)T \)
79 \( 1 + (0.576 - 0.816i)T \)
83 \( 1 + (-0.998 - 0.0455i)T \)
89 \( 1 + (-0.291 - 0.956i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−18.40393857287384694855412803730, −17.74125765234836252663597272987, −16.88242872712166521695652752993, −16.58963520924523455954947906146, −15.33321633160797767042362978113, −15.0327382725874876351673555248, −14.05721750559161040047962537297, −13.55678749338150510502370678198, −13.40396788032853744427201674099, −12.72400581547568977265913272540, −11.61625741510278234801401096808, −10.99573589400191976538201867131, −10.25547019938413184876173296896, −9.18185156580088099616942245120, −8.543365620113717162786420442185, −7.65403333430913413520061174676, −7.2109280879042106699976013385, −6.36445120552770145483471415649, −6.03429077534248459418512187652, −5.11729142236078194562180321158, −3.94124274679436880457442642960, −3.2989915497026264342529824058, −2.82074204221787941825192884370, −1.84073996202240861807596468452, −0.84811741684505669266402012961, 1.27324293688498962926057583305, 1.91339637546097117874652499626, 2.90337897230981640310501289063, 3.15275897140565528694315964038, 4.41031116162057626429447674487, 4.85184027243678203956899399455, 5.703823241893842969333042706, 6.010589401395648465544430723534, 7.277341580617900963812249055299, 8.3114554531505677612825820932, 9.00642680335543925045682194986, 9.65561758441743132068088360451, 10.15179009144274501008956499035, 10.813599872520304681787410305537, 11.767449323803772458719084625036, 12.61143366101592830854885482106, 12.9321530237294416293678852372, 13.748324772000071847071031874209, 14.39642144728340302331506508585, 14.96024110742023657585517910347, 15.71262397428054432924936951364, 16.166722102823857126006687162704, 16.89503689938641998514185567402, 18.18206091606310350302579769922, 18.51501597371852691222845722786

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