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Properties

Label 1-4031-4031.1520-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.970 - 0.240i$
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  + (−0.557 + 0.829i)2-s + (0.761 + 0.648i)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 + (0.979 + 0.203i)8-s + (0.158 + 0.987i)9-s + (0.942 − 0.334i)10-s + (−0.842 + 0.538i)11-s + (0.313 − 0.949i)12-s + (−0.983 − 0.181i)13-s + (0.942 + 0.334i)14-s + (−0.225 − 0.974i)15-s + (−0.715 + 0.699i)16-s + (−0.480 + 0.877i)17-s + ⋯
L(s)  = 1  + (−0.557 + 0.829i)2-s + (0.761 + 0.648i)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 + (0.979 + 0.203i)8-s + (0.158 + 0.987i)9-s + (0.942 − 0.334i)10-s + (−0.842 + 0.538i)11-s + (0.313 − 0.949i)12-s + (−0.983 − 0.181i)13-s + (0.942 + 0.334i)14-s + (−0.225 − 0.974i)15-s + (−0.715 + 0.699i)16-s + (−0.480 + 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (1520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5942231959 - 0.07266835591i\)
\(L(\frac12)\) \(\approx\) \(0.5942231959 - 0.07266835591i\)
\(L(1)\) \(\approx\) \(0.6304927728 + 0.2516206008i\)
\(L(1)\) \(\approx\) \(0.6304927728 + 0.2516206008i\)

Euler product

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

−18.589565014047208966590681711406, −18.11749278007227491960334547657, −17.66434213867328575533287812978, −16.175438167556582568186113288848, −15.98597362977536835009502134769, −15.06751591524533633264735826330, −14.25072065619203278430321951166, −13.68434554690493469814773774789, −12.79003702186140272146565793633, −12.24065986119541966654782132145, −11.70678338448693199054711654981, −11.08382159203870326179203352142, −10.09599144057598719284879427432, −9.4007368627452024976995403201, −8.853283118484944291062772419072, −8.02815087016092122811193132917, −7.50240774599299453065876770410, −7.01665138719245231215015321685, −5.85132810824820864106169835622, −4.84611179556181618342580188470, −3.77580883806129434611577059271, −3.13964825645538094132505231599, −2.50644793042611146210808242281, −2.081033202343462044166499879935, −0.677739746054146866322703144453, 0.26856632011226210892785118790, 1.51067498087917044893019633843, 2.48947165991473666494394000126, 3.68245528558347283435708388384, 4.25973861732585967915393387621, 4.91707797951893855717428866682, 5.564069159643464394640737927417, 6.8550285072355170948011217574, 7.63168169377356602930551998562, 7.777833340944674679847171269584, 8.613394577351667136118873064446, 9.396551871885319619671291493139, 10.01056765538601388170722038468, 10.51589711916110929944687983368, 11.28569388880132206155008024810, 12.54488355585076408554112902715, 13.08796551293183366457495985099, 13.94380046353339237589060794394, 14.50641278220768955326197853996, 15.256316959556820198508114646007, 15.84534518910786626336488392556, 16.15634944971637648910388779044, 17.07620990436768108815571089380, 17.42252237337390662991503721376, 18.5232779462638001408465245333

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