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Properties

Label 1-40e2-1600.29-r0-0-0
Degree $1$
Conductor $1600$
Sign $0.999 + 0.00392i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (0.0784 − 0.996i)3-s + (−0.707 + 0.707i)7-s + (−0.987 − 0.156i)9-s + (0.972 + 0.233i)11-s + (0.233 + 0.972i)13-s + (−0.951 − 0.309i)17-s + (−0.760 − 0.649i)19-s + (0.649 + 0.760i)21-s + (−0.156 − 0.987i)23-s + (−0.233 + 0.972i)27-s + (0.0784 − 0.996i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.852 + 0.522i)37-s + (0.987 − 0.156i)39-s + ⋯
L(s)  = 1  + (0.0784 − 0.996i)3-s + (−0.707 + 0.707i)7-s + (−0.987 − 0.156i)9-s + (0.972 + 0.233i)11-s + (0.233 + 0.972i)13-s + (−0.951 − 0.309i)17-s + (−0.760 − 0.649i)19-s + (0.649 + 0.760i)21-s + (−0.156 − 0.987i)23-s + (−0.233 + 0.972i)27-s + (0.0784 − 0.996i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.852 + 0.522i)37-s + (0.987 − 0.156i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.999 + 0.00392i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ 0.999 + 0.00392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267876006 + 0.002489471916i\)
\(L(\frac12)\) \(\approx\) \(1.267876006 + 0.002489471916i\)
\(L(1)\) \(\approx\) \(0.9704254732 - 0.1550398212i\)
\(L(1)\) \(\approx\) \(0.9704254732 - 0.1550398212i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.0784 - 0.996i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.972 + 0.233i)T \)
13 \( 1 + (0.233 + 0.972i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.760 - 0.649i)T \)
23 \( 1 + (-0.156 - 0.987i)T \)
29 \( 1 + (0.0784 - 0.996i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.852 + 0.522i)T \)
41 \( 1 + (0.987 + 0.156i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.649 + 0.760i)T \)
59 \( 1 + (0.852 + 0.522i)T \)
61 \( 1 + (-0.522 - 0.852i)T \)
67 \( 1 + (-0.649 + 0.760i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.156 + 0.987i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (0.760 + 0.649i)T \)
89 \( 1 + (0.987 - 0.156i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−20.39285247254377707148054546933, −19.76020711239382171937149798232, −19.392740038873147853486540037119, −18.0841114780467842785964673338, −17.287264679157232818785242853992, −16.68298579067497158167130421658, −16.07233413276808140789912308621, −15.2088897102543499230937225665, −14.66016727177507829273774426375, −13.719571171089562853427867493287, −13.07464260973623195997718146917, −12.11468012047032492368384816867, −11.03467478884228641198621505479, −10.65931921993675969508362147740, −9.751802192551238265555184296293, −9.137073847105594164079322169735, −8.323666645479260506122674887557, −7.35250649299480687706979108268, −6.26850107883884317351448821869, −5.73468184530584549428810728870, −4.52630688437076370262440420251, −3.79089070189553589826059154103, −3.31221659045642951497970719990, −2.04935516820494207478034996480, −0.591727069610420967281365866094, 0.87233684362568724913284728302, 2.13085809001675487777004301138, 2.564442974499671109046254328290, 3.82966130220293073524793610598, 4.726068782934608933678872175527, 6.03420490788785639525885741610, 6.55161003479736086360409405656, 7.018067682434915357630630826342, 8.284409477672533649466568334487, 8.96645157984966037190895732282, 9.43543696114870760003239051126, 10.75973211591320928210368997878, 11.66515520710205672759498487477, 12.10592530261160548831897672682, 12.98036684217077745670058789278, 13.55453449493983243017454379715, 14.43581676124680041762204640575, 15.09791645875429964593501234270, 16.08176988372793917470428819222, 16.847345856833702176758371593376, 17.569077859674823496199354908332, 18.413764693327671123789556090721, 18.92814510041275156816773961572, 19.71557180501477265749955622807, 20.0921733935587692827924460587

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