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Properties

Label 1-600-600.509-r1-0-0
Degree $1$
Conductor $600$
Sign $0.187 - 0.982i$
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 + (0.309 + 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)23-s + (−0.809 + 0.587i)29-s + (−0.809 − 0.587i)31-s + (0.309 − 0.951i)37-s + (−0.309 + 0.951i)41-s + 43-s + (−0.809 + 0.587i)47-s + 49-s + (0.809 − 0.587i)53-s + (0.309 − 0.951i)59-s + ⋯
L(s)  = 1  − 7-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 + (0.309 + 0.951i)23-s + (−0.809 + 0.587i)29-s + (−0.809 − 0.587i)31-s + (0.309 − 0.951i)37-s + (−0.309 + 0.951i)41-s + 43-s + (−0.809 + 0.587i)47-s + 49-s + (0.809 − 0.587i)53-s + (0.309 − 0.951i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(64.4789\)
Root analytic conductor: \(64.4789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 600,\ (1:\ ),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8988814460 - 0.7436194031i\)
\(L(\frac12)\) \(\approx\) \(0.8988814460 - 0.7436194031i\)
\(L(1)\) \(\approx\) \(0.9093970826 - 0.05683576846i\)
\(L(1)\) \(\approx\) \(0.9093970826 - 0.05683576846i\)

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 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
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

−22.96968334532321376527709620542, −22.17938731548560151276671909805, −21.62741885437870506832371100949, −20.54009332175664242072424464912, −19.65358529634131501082857087078, −19.00036186892139408763447209360, −18.26227827636952951981396688624, −17.04436060243191888842560093584, −16.404407081771974274121013289183, −15.698689345151019085809740854836, −14.658253749818919556398541246737, −13.62764304034151545023025778644, −13.12247133753120035031678224335, −11.98087642885147899803376694490, −11.17330552307074440732068886918, −10.26270839144008492654252605987, −9.10281702833528336709885747186, −8.7142549873578027539071357524, −7.23090723175452967043233495265, −6.48149005038813714321586149835, −5.66383980286999120590855493110, −4.29584624399532258325508279560, −3.44679769717104732965042241790, −2.36420327340981634785802737907, −0.92939711108195340673301791072, 0.339464288016158806987191232925, 1.756843940086482404764583163300, 3.03211984401910595502940408776, 3.85220380830382906879554928587, 5.123911316347583086767025682523, 6.03639195281054110077905229311, 7.07883794719166540131349714730, 7.78531678752981253863276222179, 9.2438697343783376627356846437, 9.61735798742025926342145173407, 10.72231830455610858770724775157, 11.66763916041816874386107913579, 12.754905582430562279399537499048, 13.17357932769265758021574907730, 14.32962464175026939333845838160, 15.29167070806966800774420160133, 15.95015586336943420007139660357, 16.821124099704799216418590690, 17.85975733697856865492733309743, 18.41177909504008474449733744015, 19.64861836626965919890414575537, 20.08305236193486671949481895657, 20.93373985120963661394770590911, 22.21682349935040914950647609079, 22.60407241811647542458714356480

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