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Properties

Label 1-4600-4600.459-r0-0-0
Degree $1$
Conductor $4600$
Sign $-0.876 - 0.481i$
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  + (0.809 − 0.587i)3-s − 7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + (0.309 − 0.951i)41-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s − 7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + (0.309 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4190341916 - 1.632030424i\)
\(L(\frac12)\) \(\approx\) \(0.4190341916 - 1.632030424i\)
\(L(1)\) \(\approx\) \(1.073570389 - 0.5400591988i\)
\(L(1)\) \(\approx\) \(1.073570389 - 0.5400591988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
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

−18.56146073143664870725468997232, −17.889260250113233556733956676987, −17.03866029039236896040942060359, −16.241017422134003305878495696734, −15.77378145676816014237170008624, −15.28445493499621835716728341150, −14.47032715375330555070528155240, −13.80099185018856749130558205530, −13.174010858319096826588069928464, −12.62762679344735715507452744955, −11.69114857023196074053463142862, −10.8861746383273041064572986544, −10.10785563792958822268910873984, −9.62877668347820604569053247358, −8.98695900840218339986270113972, −8.4071824234611886325683906249, −7.357502811990383398426638059590, −6.905480003148905028747700725556, −6.03935556944239877981999645965, −5.02785169181040999571134002031, −4.28405464468789475025634597879, −3.794666608485799683331732273578, −2.74731005632764494473518451696, −2.34448952694253112345215953203, −1.255039371505690213705655186517, 0.4265544088581262349774784308, 1.172316443299413673587544144, 2.39416391399447661443043992872, 3.06417806622975695510333467921, 3.41953334349099682608327288015, 4.453046049849098906326847097597, 5.58541415048230112546751204322, 6.18850043109457086856772495337, 6.88135341664716974823365072997, 7.62931732839142645316305444207, 8.38189178136495596207212766028, 8.83727373306187078607507521698, 9.75391686025015874258681796237, 10.229256452536345455835151480247, 11.17984892659982560520749203748, 12.049651374165507916262493800831, 12.612929898780906239551716576, 13.47520792373958190703972205566, 13.62923558280958155121208978589, 14.39288334592115018280349889140, 15.479517032708147609461530220309, 15.73959000038920910340213562445, 16.40124006031976253104136193126, 17.47338674142269688690890984673, 18.042163658558935109523903510787

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