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Properties

Label 1-6031-6031.2014-r0-0-0
Degree $1$
Conductor $6031$
Sign $-0.999 + 0.0105i$
Analytic cond. $28.0078$
Root an. cond. $28.0078$
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  + (0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + 6-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + 14-s + (0.173 − 0.984i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + 6-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + 14-s + (0.173 − 0.984i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6031\)    =    \(37 \cdot 163\)
Sign: $-0.999 + 0.0105i$
Analytic conductor: \(28.0078\)
Root analytic conductor: \(28.0078\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6031} (2014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6031,\ (0:\ ),\ -0.999 + 0.0105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0004180802377 + 0.07915846011i\)
\(L(\frac12)\) \(\approx\) \(0.0004180802377 + 0.07915846011i\)
\(L(1)\) \(\approx\) \(0.7592076671 - 0.06561298746i\)
\(L(1)\) \(\approx\) \(0.7592076671 - 0.06561298746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
163 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + T \)
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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

−17.30479468047214984885881681769, −16.95099111715055309228735983489, −15.85290966192283255370698498568, −15.39391524925537351836601239369, −14.80590715689276562782328874002, −14.18694395888348788015756915735, −13.43502628527899510278288497924, −12.928848382346078357060198927, −12.41240463514889188557711103731, −11.58447124680415652794494395918, −10.766209758896881008305439397553, −10.0836565072911884668325884019, −9.120741380167289352708609161177, −8.14700369705811957473524027711, −7.87811445256065984605110255337, −7.38047914839533900833585178167, −6.75629942429803361639851197122, −6.13721510332817462716617169275, −5.11604939138207555691489381119, −4.56053294291063708474330911540, −3.44412712255240176042239991277, −3.276807160169561711511019528437, −1.91754567689733438734702395637, −0.86386475030566198290172146458, −0.024555711464331081520338591407, 1.15350952593089895494695892049, 2.34046324499174948317423284509, 2.88555176709896021313855729655, 3.589632995152460949494761382303, 4.376592510626219463765433410620, 4.75218378096397418154902827080, 5.63753746843565904602673508656, 6.153153470363023163154758958602, 7.723620248561240588382876778436, 8.33138960422203081900501777260, 8.78143637866964607372839753562, 9.47530679928281142376777669751, 10.04567864144451778254069866632, 11.00322236765356090406680495032, 11.43084742686905396953476136325, 11.901625956257985578942978673140, 12.51412216194032665188432352367, 13.467260145623991281777006254228, 14.20705436950037172274473095686, 14.606677916938575208765401745373, 15.46243053148356580571230175352, 16.10035704805376188764435751989, 16.43064331984064143857425651092, 17.39871636292719974940838316424, 18.38069934016859793593696203221

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