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Properties

Label 1-4031-4031.684-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.911 + 0.411i$
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  + (0.313 − 0.949i)2-s + (0.480 − 0.877i)3-s + (−0.803 − 0.595i)4-s + (0.648 − 0.761i)5-s + (−0.682 − 0.730i)6-s + (−0.974 − 0.225i)7-s + (−0.816 + 0.576i)8-s + (−0.538 − 0.842i)9-s + (−0.519 − 0.854i)10-s + (−0.789 − 0.613i)11-s + (−0.907 + 0.419i)12-s + (−0.0227 − 0.999i)13-s + (−0.519 + 0.854i)14-s + (−0.356 − 0.934i)15-s + (0.291 + 0.956i)16-s + (0.439 + 0.898i)17-s + ⋯
L(s)  = 1  + (0.313 − 0.949i)2-s + (0.480 − 0.877i)3-s + (−0.803 − 0.595i)4-s + (0.648 − 0.761i)5-s + (−0.682 − 0.730i)6-s + (−0.974 − 0.225i)7-s + (−0.816 + 0.576i)8-s + (−0.538 − 0.842i)9-s + (−0.519 − 0.854i)10-s + (−0.789 − 0.613i)11-s + (−0.907 + 0.419i)12-s + (−0.0227 − 0.999i)13-s + (−0.519 + 0.854i)14-s + (−0.356 − 0.934i)15-s + (0.291 + 0.956i)16-s + (0.439 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.911 + 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6360138950 - 0.1367914408i\)
\(L(\frac12)\) \(\approx\) \(-0.6360138950 - 0.1367914408i\)
\(L(1)\) \(\approx\) \(0.4096541958 - 0.9345320440i\)
\(L(1)\) \(\approx\) \(0.4096541958 - 0.9345320440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.313 - 0.949i)T \)
3 \( 1 + (0.480 - 0.877i)T \)
5 \( 1 + (0.648 - 0.761i)T \)
7 \( 1 + (-0.974 - 0.225i)T \)
11 \( 1 + (-0.789 - 0.613i)T \)
13 \( 1 + (-0.0227 - 0.999i)T \)
17 \( 1 + (0.439 + 0.898i)T \)
19 \( 1 + (0.557 - 0.829i)T \)
23 \( 1 + (-0.682 + 0.730i)T \)
31 \( 1 + (-0.842 - 0.538i)T \)
37 \( 1 + (-0.356 + 0.934i)T \)
41 \( 1 + (0.699 - 0.715i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.181 + 0.983i)T \)
53 \( 1 + (-0.746 + 0.665i)T \)
59 \( 1 + (0.917 + 0.398i)T \)
61 \( 1 + (-0.699 - 0.715i)T \)
67 \( 1 + (-0.377 + 0.926i)T \)
71 \( 1 + (-0.983 - 0.181i)T \)
73 \( 1 + (-0.0909 - 0.995i)T \)
79 \( 1 + (0.269 - 0.962i)T \)
83 \( 1 + (0.377 + 0.926i)T \)
89 \( 1 + (0.987 + 0.158i)T \)
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

−18.800749769841938075868857562479, −18.4077076815565208828639407472, −17.73137525505350557052300977204, −16.65244052421606647831463800092, −16.24983041092809392840223662018, −15.85469129404209610154222002994, −14.88912833915687564181662565923, −14.47885909856254899298156715701, −13.89522805698903175328781811473, −13.2734192986607679721768167745, −12.46768039813404621939323707120, −11.62723486913174159707077479530, −10.49507210724485481170266582402, −9.859158490557316890875738199949, −9.502087413700101693591441104190, −8.79283387651621063768082881482, −7.80785783764489985829286358716, −7.16434558983308453614252641394, −6.45956693077825716705442653281, −5.63551750154420210445394894343, −5.11758522857431535714525923431, −4.1684532348600289012301238980, −3.415182513630094563708766487, −2.80876595820960776096091110253, −1.994314019131931531497119774257, 0.15939239238972028954118141623, 0.98186289778263824603781691792, 1.76013859738126884937472778856, 2.695101145014088943410431564626, 3.1960849637983501229545165261, 3.95278881088419872017256507202, 5.174719129922393933429210551442, 5.785721492115165423750247912461, 6.23870303716917484256121114717, 7.50466330178145315005385122309, 8.180243741797109234007865340872, 8.94228128778550958587835120343, 9.54025326261795807094311183908, 10.21802370669497366861216419230, 10.90443926990797039681508628626, 11.979467609016132727775208043535, 12.51730859838881219913225163423, 13.1182119690101550975342848882, 13.46454334662918336483774319453, 13.94234190351983462682383123741, 14.971007283435495542708263291097, 15.673354574331203821911264005766, 16.59163021071208211240147435417, 17.58670163795532607579577414625, 17.80342478216943969504587322899

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