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Properties

Label 1-4031-4031.829-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.896 - 0.443i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.896 - 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.264899435 - 0.7640705929i\)
\(L(\frac12)\) \(\approx\) \(3.264899435 - 0.7640705929i\)
\(L(1)\) \(\approx\) \(1.768879189 - 0.7691921220i\)
\(L(1)\) \(\approx\) \(1.768879189 - 0.7691921220i\)

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

−18.1144681066354424272649371299, −17.49697878245090792317580486844, −16.83211321580933381774607753613, −16.62710060937418132586119849451, −15.86044328615388215476160259352, −14.88985177425657657588132303875, −14.434973824591809002207276336215, −13.991067317157479679088905550762, −13.32294839870189549695534074161, −12.525481530179821936107160479284, −11.78711158070197462357759617728, −10.74994995821719127101176436602, −10.35355905364923902880204537463, −9.21991212192533697206934661109, −8.89976174319849920622027929221, −7.984409644316359366531786981173, −7.42718609672365783495488779739, −6.288180691602884801047078240354, −5.71862013057743895013210506289, −4.90895230038860149861674325860, −4.49521592522872002431240818658, −3.66946107513021073835363220644, −2.9208993681675436908913578902, −1.99622112610973534134740779524, −0.6466385151594434843251852024, 1.33034628044281889054538433875, 1.73181096321991647147089047962, 2.41198099533642737155021496670, 3.053822867084837363643419936292, 4.01843544896604305535132201611, 5.00200638440666075285732441870, 5.887217508786356588606371522870, 6.13490280090696042605440974372, 7.15084512859497820848892573744, 7.75522272434094753709131635088, 8.944588010316409575614811507066, 9.60050055351012641750583088576, 9.943267135920789898261720609143, 11.26439835960609121172224505703, 11.66011908924973589683019860030, 12.301847625453781125013895977681, 12.82808649711647118608697180695, 13.7407046053962018578782074086, 14.314777905938612901154847170453, 14.66242025810445183767515644708, 15.14656145235455349333988890193, 16.4813174873923299300772164250, 17.346166970111610527883921694799, 18.02686757072079413966776976885, 18.51807798763840446863133448146

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