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Properties

Label 1-600-600.269-r1-0-0
Degree $1$
Conductor $600$
Sign $0.968 - 0.248i$
Analytic cond. $64.4789$
Root an. cond. $64.4789$
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  − 7-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.809 + 0.587i)23-s + (0.309 − 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.809 + 0.587i)41-s + 43-s + (0.309 − 0.951i)47-s + 49-s + (−0.309 + 0.951i)53-s + (−0.809 − 0.587i)59-s + ⋯
L(s)  = 1  − 7-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.809 + 0.587i)23-s + (0.309 − 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.809 + 0.587i)41-s + 43-s + (0.309 − 0.951i)47-s + 49-s + (−0.309 + 0.951i)53-s + (−0.809 − 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(64.4789\)
Root analytic conductor: \(64.4789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 600,\ (1:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.108578059 - 0.1400459771i\)
\(L(\frac12)\) \(\approx\) \(1.108578059 - 0.1400459771i\)
\(L(1)\) \(\approx\) \(0.8264971252 + 0.002912887514i\)
\(L(1)\) \(\approx\) \(0.8264971252 + 0.002912887514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
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

−22.72314698994895916714715157468, −22.30437718404177176960186800830, −21.2090636054226891812841625238, −20.54199128516409271303909906573, −19.44735381971103820364845989242, −18.87360501740366650049328745423, −18.10613388125835400928984761501, −16.86091996675860810993773686961, −16.27664474820924647479887757124, −15.584712392198408668104186395619, −14.349309059061150709925656891144, −13.747554378520317838743966047133, −12.6287608418089493629093549034, −12.100579007196019252643894262339, −10.872946665722753494770783723274, −10.0109575663997698858422026901, −9.29697240990887004092781436077, −8.17448114079647768749927268377, −7.24739985072744006663437388556, −6.27967029164376978635412518430, −5.39011303684923644002645973702, −4.232560006708005241495038385763, −3.11742507118741484932096804698, −2.25183619038890550600700826857, −0.57805517389159775929785924364, 0.46940113459253445880425061681, 2.14578926680705335575617246822, 3.02965529107835607552659005434, 4.171119896423925100017428079639, 5.27683971606099859855921899930, 6.20344054004657530477663375993, 7.22808359219366544235365715682, 8.03035008551200307048572890666, 9.212559043287099114126906698777, 10.07579401141251901741280644939, 10.64908578078664265421133275278, 12.05234755973980448585376668564, 12.70281811682653838361772960744, 13.401046582330605418929996928770, 14.51266599557938934866378656587, 15.53489973312672347956817133977, 15.91269687792117772803714323065, 17.25024246015090042564268530598, 17.65740614633815063529235688584, 18.88160882577624852728475159702, 19.57780314062594315305490571724, 20.201169523152662856720797213401, 21.37966004302738960651706660152, 21.97181869040172743089427428984, 22.976201616165066298422477521254

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