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Properties

Label 1-4031-4031.1768-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.998 + 0.0511i$
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  + (0.0682 − 0.997i)2-s + (0.334 + 0.942i)3-s + (−0.990 − 0.136i)4-s + (−0.917 + 0.398i)5-s + (0.962 − 0.269i)6-s + (0.962 + 0.269i)7-s + (−0.203 + 0.979i)8-s + (−0.775 + 0.631i)9-s + (0.334 + 0.942i)10-s + (−0.460 + 0.887i)11-s + (−0.203 − 0.979i)12-s + (−0.334 − 0.942i)13-s + (0.334 − 0.942i)14-s + (−0.682 − 0.730i)15-s + (0.962 + 0.269i)16-s + (−0.854 + 0.519i)17-s + ⋯
L(s)  = 1  + (0.0682 − 0.997i)2-s + (0.334 + 0.942i)3-s + (−0.990 − 0.136i)4-s + (−0.917 + 0.398i)5-s + (0.962 − 0.269i)6-s + (0.962 + 0.269i)7-s + (−0.203 + 0.979i)8-s + (−0.775 + 0.631i)9-s + (0.334 + 0.942i)10-s + (−0.460 + 0.887i)11-s + (−0.203 − 0.979i)12-s + (−0.334 − 0.942i)13-s + (0.334 − 0.942i)14-s + (−0.682 − 0.730i)15-s + (0.962 + 0.269i)16-s + (−0.854 + 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.998 + 0.0511i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (1768, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.998 + 0.0511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01263026799 + 0.4931402684i\)
\(L(\frac12)\) \(\approx\) \(0.01263026799 + 0.4931402684i\)
\(L(1)\) \(\approx\) \(0.8052451334 + 0.09819124833i\)
\(L(1)\) \(\approx\) \(0.8052451334 + 0.09819124833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.0682 - 0.997i)T \)
3 \( 1 + (0.334 + 0.942i)T \)
5 \( 1 + (-0.917 + 0.398i)T \)
7 \( 1 + (0.962 + 0.269i)T \)
11 \( 1 + (-0.460 + 0.887i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (-0.854 + 0.519i)T \)
19 \( 1 + (-0.854 + 0.519i)T \)
23 \( 1 + (0.962 + 0.269i)T \)
31 \( 1 + (0.775 + 0.631i)T \)
37 \( 1 + (-0.682 + 0.730i)T \)
41 \( 1 + (0.576 - 0.816i)T \)
43 \( 1 - T \)
47 \( 1 + (0.917 + 0.398i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (-0.990 + 0.136i)T \)
61 \( 1 + (0.576 + 0.816i)T \)
67 \( 1 + (0.460 + 0.887i)T \)
71 \( 1 + (-0.917 + 0.398i)T \)
73 \( 1 + (-0.203 + 0.979i)T \)
79 \( 1 + (0.576 + 0.816i)T \)
83 \( 1 + (0.460 - 0.887i)T \)
89 \( 1 + (-0.682 - 0.730i)T \)
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

−18.06671200445744228809548073807, −17.38710970355904519343725937481, −16.79373917625321420412016831055, −16.175231115862495953390162690269, −15.178266024877310317684131930352, −14.91367031503336588424802669135, −13.96269093772448938089178359346, −13.54841036821423530605358311789, −12.87664154560952578650027728759, −12.07698858449647531905922576537, −11.3727041833074389409151594170, −10.78819755525032951271665538424, −9.25385235051940178697436425189, −8.83276274683706199322793709909, −8.16173810432644781911863205489, −7.743944563580406808518923409534, −6.887745986641147566159862962729, −6.5253452802259222765296350639, −5.33821593496849380211746338512, −4.714646747935787850464918060867, −4.05933958792077344619607053340, −3.08340889876542431423130870102, −2.06439955777053934145979905066, −0.91399954385266782682156420684, −0.15757057509862680523038432762, 1.391254456583602273997423001196, 2.46313583463714890517833738841, 2.85329538188585648136162736667, 3.867084622618689859352613925086, 4.441283169449711108593287804957, 4.95183219253397999650785169950, 5.723904066743703516716629065426, 7.129784432009915297534523903, 8.02914966907133114514001085140, 8.44893141888287077508838679136, 9.09731434532838793414738857079, 10.23756930084491626707064567416, 10.50623486348967472696051643310, 11.07555361335749978889094048330, 11.84976182424975679481740953243, 12.46281576030144649852845062945, 13.236449706040821413487053005241, 14.20184515177336375686617156680, 14.80274986183549009596066136344, 15.29991486008337535215261657161, 15.65517038686245553842680812870, 17.12755595515421843919806358084, 17.39146064765842709766211153332, 18.248918301120049040495810317150, 19.02881250798382829556572844064

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