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Properties

Label 1-4600-4600.2259-r0-0-0
Degree $1$
Conductor $4600$
Sign $0.981 - 0.192i$
Analytic cond. $21.3623$
Root an. cond. $21.3623$
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  + (−0.696 − 0.717i)3-s + (0.654 − 0.755i)7-s + (−0.0285 + 0.999i)9-s + (−0.774 − 0.633i)11-s + (−0.921 + 0.389i)13-s + (−0.870 + 0.491i)17-s + (−0.198 + 0.980i)19-s + (−0.998 + 0.0570i)21-s + (0.736 − 0.676i)27-s + (−0.198 − 0.980i)29-s + (−0.696 + 0.717i)31-s + (0.0855 + 0.996i)33-s + (0.0285 − 0.999i)37-s + (0.921 + 0.389i)39-s + (−0.564 + 0.825i)41-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)3-s + (0.654 − 0.755i)7-s + (−0.0285 + 0.999i)9-s + (−0.774 − 0.633i)11-s + (−0.921 + 0.389i)13-s + (−0.870 + 0.491i)17-s + (−0.198 + 0.980i)19-s + (−0.998 + 0.0570i)21-s + (0.736 − 0.676i)27-s + (−0.198 − 0.980i)29-s + (−0.696 + 0.717i)31-s + (0.0855 + 0.996i)33-s + (0.0285 − 0.999i)37-s + (0.921 + 0.389i)39-s + (−0.564 + 0.825i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (2259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ 0.981 - 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7621326935 - 0.07413269622i\)
\(L(\frac12)\) \(\approx\) \(0.7621326935 - 0.07413269622i\)
\(L(1)\) \(\approx\) \(0.6966514949 - 0.1757739221i\)
\(L(1)\) \(\approx\) \(0.6966514949 - 0.1757739221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.696 - 0.717i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (-0.774 - 0.633i)T \)
13 \( 1 + (-0.921 + 0.389i)T \)
17 \( 1 + (-0.870 + 0.491i)T \)
19 \( 1 + (-0.198 + 0.980i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (-0.696 + 0.717i)T \)
37 \( 1 + (0.0285 - 0.999i)T \)
41 \( 1 + (-0.564 + 0.825i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.0855 + 0.996i)T \)
59 \( 1 + (-0.921 + 0.389i)T \)
61 \( 1 + (-0.985 + 0.170i)T \)
67 \( 1 + (-0.362 - 0.931i)T \)
71 \( 1 + (0.998 - 0.0570i)T \)
73 \( 1 + (0.736 - 0.676i)T \)
79 \( 1 + (0.974 + 0.226i)T \)
83 \( 1 + (0.941 + 0.336i)T \)
89 \( 1 + (-0.897 + 0.441i)T \)
97 \( 1 + (0.941 - 0.336i)T \)
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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

−18.17353209787299284653989505254, −17.52242983534726553895089258396, −17.030500645693594372483674334946, −16.11226322607149323692356159996, −15.48210566975044198652339083398, −15.04519706053144186941152234400, −14.540611285136058740987800621057, −13.38171546584943201751179317545, −12.701067646357750440983427997923, −12.07171155373665901075246491474, −11.33736348277140136256192935275, −10.88777542076850205351659977966, −10.05738595215911842123596400142, −9.42106801198394604755924186092, −8.80069453332469373676210055764, −7.915925580921021317630599995167, −7.09380188067607331959324802665, −6.40277247678356544626342208471, −5.36856744909002492331764459341, −4.94405062881596202462002176293, −4.568039472494581082094935563383, −3.34971749169242402439902396641, −2.54017223412242076220330036493, −1.80325233174440829056845987705, −0.37220687346728149407374793962, 0.565207726308941802641862910063, 1.71699241012420534607971716346, 2.14001609656153662880829224566, 3.31178076506947351260252366833, 4.34600978016446612394797172063, 4.90141996409804918638951887053, 5.71630994575710190489689819544, 6.388493286388095826427539553566, 7.191285028800472197431888208413, 7.81021565713585296117178787582, 8.26853104527260991971856452537, 9.31289915953546547893714664108, 10.33724970233680648464907266459, 10.78949084023271751924372860145, 11.32018657319777667769890268967, 12.2039441176640401140596006154, 12.6715903415374846565808959218, 13.60578042814811780808737693958, 13.88505219765939015636385129174, 14.77648908674125704855967758827, 15.5415681198817012023227014271, 16.59734050461655340046319536278, 16.76579505204252243518732076406, 17.5375254901340037187095936750, 18.17076559169447082902913311247

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