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Properties

Label 1-4031-4031.104-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.772 + 0.634i$
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  + (0.0909 + 0.995i)2-s + (−0.557 − 0.829i)3-s + (−0.983 + 0.181i)4-s + (−0.0227 + 0.999i)5-s + (0.775 − 0.631i)6-s + (−0.158 + 0.987i)7-s + (−0.269 − 0.962i)8-s + (−0.377 + 0.926i)9-s + (−0.997 + 0.0682i)10-s + (−0.595 − 0.803i)11-s + (0.699 + 0.715i)12-s + (−0.898 − 0.439i)13-s + (−0.997 − 0.0682i)14-s + (0.842 − 0.538i)15-s + (0.934 − 0.356i)16-s + (−0.313 − 0.949i)17-s + ⋯
L(s)  = 1  + (0.0909 + 0.995i)2-s + (−0.557 − 0.829i)3-s + (−0.983 + 0.181i)4-s + (−0.0227 + 0.999i)5-s + (0.775 − 0.631i)6-s + (−0.158 + 0.987i)7-s + (−0.269 − 0.962i)8-s + (−0.377 + 0.926i)9-s + (−0.997 + 0.0682i)10-s + (−0.595 − 0.803i)11-s + (0.699 + 0.715i)12-s + (−0.898 − 0.439i)13-s + (−0.997 − 0.0682i)14-s + (0.842 − 0.538i)15-s + (0.934 − 0.356i)16-s + (−0.313 − 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.772 + 0.634i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.772 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2293374006 + 0.6409295585i\)
\(L(\frac12)\) \(\approx\) \(0.2293374006 + 0.6409295585i\)
\(L(1)\) \(\approx\) \(0.5971456674 + 0.3291212072i\)
\(L(1)\) \(\approx\) \(0.5971456674 + 0.3291212072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.0909 + 0.995i)T \)
3 \( 1 + (-0.557 - 0.829i)T \)
5 \( 1 + (-0.0227 + 0.999i)T \)
7 \( 1 + (-0.158 + 0.987i)T \)
11 \( 1 + (-0.595 - 0.803i)T \)
13 \( 1 + (-0.898 - 0.439i)T \)
17 \( 1 + (-0.313 - 0.949i)T \)
19 \( 1 + (-0.665 + 0.746i)T \)
23 \( 1 + (0.775 + 0.631i)T \)
31 \( 1 + (0.926 - 0.377i)T \)
37 \( 1 + (0.842 + 0.538i)T \)
41 \( 1 + (0.225 - 0.974i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.480 - 0.877i)T \)
53 \( 1 + (0.419 - 0.907i)T \)
59 \( 1 + (0.334 + 0.942i)T \)
61 \( 1 + (-0.225 - 0.974i)T \)
67 \( 1 + (-0.113 + 0.993i)T \)
71 \( 1 + (0.877 - 0.480i)T \)
73 \( 1 + (0.968 + 0.247i)T \)
79 \( 1 + (0.730 - 0.682i)T \)
83 \( 1 + (0.113 + 0.993i)T \)
89 \( 1 + (0.0455 + 0.998i)T \)
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

−18.00537504420819676778678240444, −17.414874244591592696423587167072, −16.97284924621921760974793539037, −16.45270899973882955670932767695, −15.36782160229038046796541909182, −14.87822668287805210327061310562, −13.99522266322439743150598569157, −13.08005623905500172778721305030, −12.67552417477917012790348149077, −12.09529079553693521092012915817, −11.11761292092156231718597610241, −10.73172644589424688743264347058, −9.86264682245035277949053006028, −9.580185599663295130362797020831, −8.71748765376951908505386373279, −7.98680892839356772048611009786, −6.8889923668879936912228150575, −5.98083857774750625847252007977, −4.94273001148708075002473440135, −4.473363794672464149141000152238, −4.29840267491842366888573909716, −3.14229812529988908890737937152, −2.23880223672095711140495647992, −1.19109273142826266288703721014, −0.32604847587032630942474553344, 0.68630706441196097063266343923, 2.267644715286672057258215482618, 2.76082073903668895617436724604, 3.70943987416537169800272979723, 5.13399253274759562532107467212, 5.33792540522649541269061556232, 6.24584731800546061258238607728, 6.67222125349321201264060521797, 7.485517644477097526553539552323, 8.018347101386817068136016233191, 8.73266015829087289631107326677, 9.72552436456695686126888481168, 10.41326899993682319256803333081, 11.33721908644139466107816956058, 11.97553293161760086328534107434, 12.66424128927640864183249527737, 13.48645711749812409351735973680, 13.853874514561068972709945482, 14.875307501697583197392776521044, 15.225232805903136590226878744119, 16.012496594078866780797915548802, 16.7476004950773142877688801783, 17.417183728671776769357179893327, 18.101983865639970780146938894973, 18.58190477543446832306507593502

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