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Properties

Label 1-40e2-1600.1109-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.726 - 0.687i$
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  + (−0.522 + 0.852i)3-s + (−0.707 − 0.707i)7-s + (−0.453 − 0.891i)9-s + (−0.0784 + 0.996i)11-s + (−0.996 + 0.0784i)13-s + (0.587 − 0.809i)17-s + (−0.233 + 0.972i)19-s + (0.972 − 0.233i)21-s + (0.891 + 0.453i)23-s + (0.996 + 0.0784i)27-s + (−0.522 + 0.852i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.760 − 0.649i)37-s + (0.453 − 0.891i)39-s + ⋯
L(s)  = 1  + (−0.522 + 0.852i)3-s + (−0.707 − 0.707i)7-s + (−0.453 − 0.891i)9-s + (−0.0784 + 0.996i)11-s + (−0.996 + 0.0784i)13-s + (0.587 − 0.809i)17-s + (−0.233 + 0.972i)19-s + (0.972 − 0.233i)21-s + (0.891 + 0.453i)23-s + (0.996 + 0.0784i)27-s + (−0.522 + 0.852i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.760 − 0.649i)37-s + (0.453 − 0.891i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01017885419 + 0.02556209655i\)
\(L(\frac12)\) \(\approx\) \(0.01017885419 + 0.02556209655i\)
\(L(1)\) \(\approx\) \(0.6364025381 + 0.1665097714i\)
\(L(1)\) \(\approx\) \(0.6364025381 + 0.1665097714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.522 + 0.852i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.0784 + 0.996i)T \)
13 \( 1 + (-0.996 + 0.0784i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.233 + 0.972i)T \)
23 \( 1 + (0.891 + 0.453i)T \)
29 \( 1 + (-0.522 + 0.852i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.760 - 0.649i)T \)
41 \( 1 + (0.453 + 0.891i)T \)
43 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.972 - 0.233i)T \)
59 \( 1 + (-0.760 - 0.649i)T \)
61 \( 1 + (-0.649 - 0.760i)T \)
67 \( 1 + (-0.972 - 0.233i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 + (-0.891 - 0.453i)T \)
79 \( 1 + (-0.587 - 0.809i)T \)
83 \( 1 + (0.233 - 0.972i)T \)
89 \( 1 + (0.453 - 0.891i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−19.53138658745483850249235878639, −19.188160736790142080195437716014, −18.78934686778592028228509884593, −17.717897425236297123952384373204, −17.05312117970687401414653048597, −16.51409595899505453087640345218, −15.516942155016843131137100520836, −14.82638654115714508634847206551, −13.74832857593616262030412875652, −13.157730044275180272640815603841, −12.412441537810611082441561003501, −11.85680974421211566343203973722, −10.98133782630933641323548775945, −10.21693825554751199265166779691, −9.14608943436445631030759498367, −8.44260716307435545532620009739, −7.5345653702385688044850194861, −6.74176078919896251202618955752, −5.92366362476228380864831284099, −5.439312008318171662758445368344, −4.306354169537861252614730000967, −2.89904323404905469556602676431, −2.488720717010383305583932949333, −1.140513181859052323263842035050, −0.012015891665354912204265706918, 1.39131871114715214751105061713, 2.841172795204945133138743097054, 3.56822141059028217126253451010, 4.55660511989170770593626614150, 5.0999219951341923087224170596, 6.106080459649698703824585284867, 7.05877884731172034027318039561, 7.583257221558074592584966027595, 9.01215560999957792413386546500, 9.66944835249895919257457138785, 10.19549195799118511313022536845, 10.86686441992818845499531296997, 12.000055364887660002471086482958, 12.40047495784365664168928882374, 13.37446128054062788844084625745, 14.506508213655593807727040115617, 14.83793506041201883731544353574, 15.94785037042844715351906423945, 16.39069946208196907895533165243, 17.17741566542533856004425497366, 17.67478247885386252423728220419, 18.71259845990090270496214441171, 19.619332913196702067977729706657, 20.25532350069358541918150480842, 20.96786333919617675422873178338

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