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Properties

Label 1-600-600.221-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: $\chi_{600} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 600,\ (1:\ ),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.311488893 + 0.2920089552i\)
\(L(\frac12)\) \(\approx\) \(2.311488893 + 0.2920089552i\)
\(L(1)\) \(\approx\) \(1.246202717 + 0.03135498902i\)
\(L(1)\) \(\approx\) \(1.246202717 + 0.03135498902i\)

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

−23.12445072243130229432718531978, −21.921649822965266879922002394958, −21.0244233292537211056510169975, −20.64387947162888014701986084193, −19.58863421996780515534148075125, −18.50443322720245366890425208371, −17.971973937875994283824282359, −17.15554267838246911601351819915, −16.05760835429374986782508286264, −15.35143762972062303187836938203, −14.45579335268780724537073906485, −13.57589919335784362825626057694, −12.80782116418216629838145913925, −11.553357274095288540686358223307, −11.11404719074661478211902398317, −10.03893587940867505655617538227, −8.970478407045486556483308482841, −8.17916177938927047654365257344, −7.21792871109543317884583020802, −6.29638208419995203961755346178, −4.87674586731806609281405734644, −4.57085411355962944357615623471, −2.96564672200737315212700744256, −1.99771168641259606647177275030, −0.74075941529510455489706418966, 0.89080902455110939586206672312, 1.976166733786587916786044967055, 3.24366661926439049673327862792, 4.26942563872808397572890046966, 5.41934068634842723932949559644, 6.07097757551850453257749924363, 7.49291554055965915692778724728, 8.22089875601088773342274095289, 8.90733187981480465296147588749, 10.40571135341014613109261545770, 10.83652228313226024282749653685, 11.779631510224743600786385871341, 12.92397321709329618610904706754, 13.54122591466979386227700343905, 14.65565987390041093711287415480, 15.263883723370916764649554524922, 16.259273598599436442012064611121, 17.14247341255757419608545494975, 18.03712074354945242109392152708, 18.62146825040061621869037344486, 19.63103035400705235825335893735, 20.65736761810775228744226332565, 21.18650506534670911871565665100, 21.92947631671537025847848144886, 23.203750342044189285779427903096

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