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Properties

Label 1-40e2-1600.1069-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.829 - 0.558i$
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  + (−0.760 − 0.649i)3-s + (0.707 − 0.707i)7-s + (0.156 + 0.987i)9-s + (−0.852 − 0.522i)11-s + (−0.522 − 0.852i)13-s + (0.951 − 0.309i)17-s + (−0.0784 + 0.996i)19-s + (−0.996 + 0.0784i)21-s + (0.987 + 0.156i)23-s + (0.522 − 0.852i)27-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.972 + 0.233i)37-s + (−0.156 + 0.987i)39-s + ⋯
L(s)  = 1  + (−0.760 − 0.649i)3-s + (0.707 − 0.707i)7-s + (0.156 + 0.987i)9-s + (−0.852 − 0.522i)11-s + (−0.522 − 0.852i)13-s + (0.951 − 0.309i)17-s + (−0.0784 + 0.996i)19-s + (−0.996 + 0.0784i)21-s + (0.987 + 0.156i)23-s + (0.522 − 0.852i)27-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.972 + 0.233i)37-s + (−0.156 + 0.987i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.829 - 0.558i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.829 - 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2666073127 - 0.8727141511i\)
\(L(\frac12)\) \(\approx\) \(0.2666073127 - 0.8727141511i\)
\(L(1)\) \(\approx\) \(0.7419067313 - 0.3532012337i\)
\(L(1)\) \(\approx\) \(0.7419067313 - 0.3532012337i\)

Euler product

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

−21.05937466553930875540688046063, −20.22711711096053855176575889492, −19.150318273298440942321274226057, −18.37369100548763800400418120204, −17.79139755497115696586408442761, −17.02895420032935114864335911241, −16.34274458325899784683562874776, −15.54148218178202892585923844976, −14.84380781703109688452035433065, −14.37433308583505590607816298369, −12.94518785765199680647714067118, −12.44409799150821464656590791804, −11.50175218088351258485352707655, −11.05688603550384682818653064397, −10.140442412739766257670769408942, −9.369640140856710083583813758627, −8.68093200516699261316133036285, −7.55959861048720919031594107896, −6.80135588346784086821002581999, −5.74424620698571981393469006526, −4.98316306022434025521229588062, −4.62881506838750321393069087555, −3.35573521428985480328593335126, −2.363879387737659837705750916325, −1.250304354033102478225439336176, 0.41201745792413161208143466524, 1.32205136928474871777955678251, 2.39364269653174029572754022206, 3.46083489331559417790938447600, 4.65120462693476366201930990911, 5.41111770927538228877265808105, 5.95044798192019926639794535743, 7.178409193694421699111358408413, 7.79482562476043222906824362081, 8.17278592451460776638761667946, 9.730179063321785665407100451808, 10.394477872182741369341866300597, 11.157262722128790314759072640825, 11.69658232373881602354580734505, 12.83814380966699967652260582954, 13.1069146517470802206269893789, 14.13991830259500496389470261543, 14.79749142546979619680519303555, 15.85212983196459276973640261573, 16.732985845722966102562040334251, 17.123395838572425827798297839189, 17.93692516753898094863977629665, 18.67676529873520528060505518413, 19.144260160132011279124117639923, 20.29855178202891753046077124386

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