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Properties

Label 1-4031-4031.12-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.581 - 0.813i$
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  + (−0.926 − 0.377i)2-s + (0.987 + 0.158i)3-s + (0.715 + 0.699i)4-s + (−0.291 + 0.956i)5-s + (−0.854 − 0.519i)6-s + (−0.877 − 0.480i)7-s + (−0.398 − 0.917i)8-s + (0.949 + 0.313i)9-s + (0.631 − 0.775i)10-s + (0.907 − 0.419i)11-s + (0.595 + 0.803i)12-s + (−0.934 + 0.356i)13-s + (0.631 + 0.775i)14-s + (−0.439 + 0.898i)15-s + (0.0227 + 0.999i)16-s + (−0.842 − 0.538i)17-s + ⋯
L(s)  = 1  + (−0.926 − 0.377i)2-s + (0.987 + 0.158i)3-s + (0.715 + 0.699i)4-s + (−0.291 + 0.956i)5-s + (−0.854 − 0.519i)6-s + (−0.877 − 0.480i)7-s + (−0.398 − 0.917i)8-s + (0.949 + 0.313i)9-s + (0.631 − 0.775i)10-s + (0.907 − 0.419i)11-s + (0.595 + 0.803i)12-s + (−0.934 + 0.356i)13-s + (0.631 + 0.775i)14-s + (−0.439 + 0.898i)15-s + (0.0227 + 0.999i)16-s + (−0.842 − 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.581 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8377737527 - 0.4307268660i\)
\(L(\frac12)\) \(\approx\) \(0.8377737527 - 0.4307268660i\)
\(L(1)\) \(\approx\) \(0.7909554249 + 0.02404143745i\)
\(L(1)\) \(\approx\) \(0.7909554249 + 0.02404143745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.926 - 0.377i)T \)
3 \( 1 + (0.987 + 0.158i)T \)
5 \( 1 + (-0.291 + 0.956i)T \)
7 \( 1 + (-0.877 - 0.480i)T \)
11 \( 1 + (0.907 - 0.419i)T \)
13 \( 1 + (-0.934 + 0.356i)T \)
17 \( 1 + (-0.842 - 0.538i)T \)
19 \( 1 + (-0.0455 + 0.998i)T \)
23 \( 1 + (-0.854 + 0.519i)T \)
31 \( 1 + (-0.313 - 0.949i)T \)
37 \( 1 + (-0.439 - 0.898i)T \)
41 \( 1 + (-0.181 - 0.983i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.225 + 0.974i)T \)
53 \( 1 + (-0.613 - 0.789i)T \)
59 \( 1 + (-0.962 - 0.269i)T \)
61 \( 1 + (0.181 - 0.983i)T \)
67 \( 1 + (-0.995 + 0.0909i)T \)
71 \( 1 + (0.974 - 0.225i)T \)
73 \( 1 + (0.993 + 0.113i)T \)
79 \( 1 + (0.942 - 0.334i)T \)
83 \( 1 + (0.995 + 0.0909i)T \)
89 \( 1 + (-0.557 - 0.829i)T \)
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

−18.6321881253469401777949188574, −17.88758425424288301743753622586, −17.1987514767050898141901314681, −16.55646614617602027416135805556, −15.77939189071389088024426400266, −15.30599843448160870987556111898, −14.80746849253351271447985279526, −13.86144583867468836640973822252, −13.10453258744192157844787830493, −12.23660198126714013531571138542, −12.02893765525648055113584422730, −10.72850651626865078581190545875, −9.88913882702256008085700643692, −9.344254194264488291517093558459, −8.872369470980444162234438749541, −8.33257137554124449734297549207, −7.5114988418501078289210139469, −6.78788227883338303702478226693, −6.250845862792234904161841607078, −5.08107506631433797972967376678, −4.37102682819064110332322017185, −3.3555798137873168775012102180, −2.429999480792855885524910062344, −1.81188476968167737692782737497, −0.81373310775101251849923244455, 0.37701773414695163627867693807, 1.82023938538112064175782603426, 2.32823134735532903574900388430, 3.28725375658704581981955777679, 3.68563081739467357854202197969, 4.36113254698997538662306658749, 6.11982320307260362050659479517, 6.66533559897173265678790295851, 7.4799514607075933372690781684, 7.73765926238563035987693893128, 8.820894331786648729341667181999, 9.56579105435967544125501169377, 9.76703215209923477389348291827, 10.67147871523239148113324907935, 11.25919689171868902880077724137, 12.15971016008353816182744064842, 12.7391655150500857042167081822, 13.85671560259695537871082908039, 14.16879908454979202696425062002, 15.05058527944642885164849586982, 15.79657983471620740067384630510, 16.26074435788785737502629187593, 17.055740688734490296770230478949, 17.8191701858866493249129882195, 18.65785080623520104270875610521

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