↑

Properties

Label 1-4024-4024.1237-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.590 + 0.807i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0312 − 0.999i)3-s + (0.915 + 0.401i)5-s + (−0.570 − 0.821i)7-s + (−0.998 − 0.0625i)9-s + (−0.0687 + 0.997i)11-s + (−0.580 − 0.814i)13-s + (0.429 − 0.902i)15-s + (−0.943 − 0.331i)17-s + (−0.877 − 0.479i)19-s + (−0.838 + 0.544i)21-s + (0.217 + 0.976i)23-s + (0.677 + 0.735i)25-s + (−0.0937 + 0.995i)27-s + (0.253 − 0.967i)29-s + (−0.971 − 0.235i)31-s + ⋯
L(s)  = 1  + (0.0312 − 0.999i)3-s + (0.915 + 0.401i)5-s + (−0.570 − 0.821i)7-s + (−0.998 − 0.0625i)9-s + (−0.0687 + 0.997i)11-s + (−0.580 − 0.814i)13-s + (0.429 − 0.902i)15-s + (−0.943 − 0.331i)17-s + (−0.877 − 0.479i)19-s + (−0.838 + 0.544i)21-s + (0.217 + 0.976i)23-s + (0.677 + 0.735i)25-s + (−0.0937 + 0.995i)27-s + (0.253 − 0.967i)29-s + (−0.971 − 0.235i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.590 + 0.807i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.590 + 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6626140735 + 0.3362800384i\)
\(L(\frac12)\) \(\approx\) \(0.6626140735 + 0.3362800384i\)
\(L(1)\) \(\approx\) \(0.8783440164 - 0.2361102155i\)
\(L(1)\) \(\approx\) \(0.8783440164 - 0.2361102155i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.0312 - 0.999i)T \)
5 \( 1 + (0.915 + 0.401i)T \)
7 \( 1 + (-0.570 - 0.821i)T \)
11 \( 1 + (-0.0687 + 0.997i)T \)
13 \( 1 + (-0.580 - 0.814i)T \)
17 \( 1 + (-0.943 - 0.331i)T \)
19 \( 1 + (-0.877 - 0.479i)T \)
23 \( 1 + (0.217 + 0.976i)T \)
29 \( 1 + (0.253 - 0.967i)T \)
31 \( 1 + (-0.971 - 0.235i)T \)
37 \( 1 + (-0.764 + 0.644i)T \)
41 \( 1 + (0.947 - 0.319i)T \)
43 \( 1 + (0.528 - 0.848i)T \)
47 \( 1 + (0.474 + 0.880i)T \)
53 \( 1 + (-0.877 + 0.479i)T \)
59 \( 1 + (0.301 + 0.953i)T \)
61 \( 1 + (-0.994 + 0.0999i)T \)
67 \( 1 + (-0.429 - 0.902i)T \)
71 \( 1 + (0.731 - 0.682i)T \)
73 \( 1 + (0.407 + 0.913i)T \)
79 \( 1 + (0.0437 - 0.999i)T \)
83 \( 1 + (-0.980 - 0.198i)T \)
89 \( 1 + (0.999 + 0.0250i)T \)
97 \( 1 + (0.168 + 0.985i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.328577196690596304759474637240, −17.57579643967162381639640631886, −16.76461778182676363755402338679, −16.388159568339100043374836459961, −15.85161494702136399572096175502, −14.88712021724526167243870255789, −14.37481446864873365227840452600, −13.76646521670563956499169346447, −12.685474231248198069882033978161, −12.47499926937033478371425608564, −11.21532770651947719903714967658, −10.80999376293885427327679645810, −9.96052806901233689367297154110, −9.31694387393827309100450926873, −8.7196118804023286259078058713, −8.454282675440325665970465854106, −6.882569020855337235685211417055, −6.18228949755064929765179965575, −5.66749852527256772409834112804, −4.885869063787370144687816013610, −4.20014082804617102113721938270, −3.23965934404457423536881315798, −2.46976635920430015282865653492, −1.817446547519231255948180841390, −0.21598136654738256262451803372, 0.922320745859226081499902303399, 1.98216762857592230000105204076, 2.46550273680680265334004878747, 3.2769847152472961159963176488, 4.38108957722395391660135328916, 5.25434360664017551888925394862, 6.113679490507089969752628080125, 6.67051121741996210198165142541, 7.37824774150470011473344306247, 7.72770254058613636122087281817, 9.095166240976178438889767022119, 9.41586263811118864931101197118, 10.48101854939343451696613818409, 10.78352209893524261074938458740, 11.84671863924197374140827061358, 12.69644176762610061531159383211, 13.11153379369664900673079594613, 13.64863408160437483896854664708, 14.30976529884563497977539282908, 15.11430529082755563870918834294, 15.71688409306107324079869171266, 17.068917060091252037573925223111, 17.30198382475816701555212347560, 17.71241458191443846949172462421, 18.51120791138743144241972039134

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