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Properties

Label 1-4012-4012.423-r0-0-0
Degree $1$
Conductor $4012$
Sign $0.0504 - 0.998i$
Analytic cond. $18.6316$
Root an. cond. $18.6316$
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  + (0.842 − 0.538i)3-s + (−0.812 − 0.583i)5-s + (0.293 − 0.955i)7-s + (0.419 − 0.907i)9-s + (0.626 + 0.779i)11-s + (0.907 − 0.419i)13-s + (−0.998 − 0.0541i)15-s + (−0.515 + 0.856i)19-s + (−0.267 − 0.963i)21-s + (0.918 − 0.395i)23-s + (0.319 + 0.947i)25-s + (−0.135 − 0.990i)27-s + (−0.0270 + 0.999i)29-s + (0.241 − 0.970i)31-s + (0.947 + 0.319i)33-s + ⋯
L(s)  = 1  + (0.842 − 0.538i)3-s + (−0.812 − 0.583i)5-s + (0.293 − 0.955i)7-s + (0.419 − 0.907i)9-s + (0.626 + 0.779i)11-s + (0.907 − 0.419i)13-s + (−0.998 − 0.0541i)15-s + (−0.515 + 0.856i)19-s + (−0.267 − 0.963i)21-s + (0.918 − 0.395i)23-s + (0.319 + 0.947i)25-s + (−0.135 − 0.990i)27-s + (−0.0270 + 0.999i)29-s + (0.241 − 0.970i)31-s + (0.947 + 0.319i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0504 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0504 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $0.0504 - 0.998i$
Analytic conductor: \(18.6316\)
Root analytic conductor: \(18.6316\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4012} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4012,\ (0:\ ),\ 0.0504 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860267500 - 1.768647728i\)
\(L(\frac12)\) \(\approx\) \(1.860267500 - 1.768647728i\)
\(L(1)\) \(\approx\) \(1.343570442 - 0.5821667902i\)
\(L(1)\) \(\approx\) \(1.343570442 - 0.5821667902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good3 \( 1 + (0.842 - 0.538i)T \)
5 \( 1 + (-0.812 - 0.583i)T \)
7 \( 1 + (0.293 - 0.955i)T \)
11 \( 1 + (0.626 + 0.779i)T \)
13 \( 1 + (0.907 - 0.419i)T \)
19 \( 1 + (-0.515 + 0.856i)T \)
23 \( 1 + (0.918 - 0.395i)T \)
29 \( 1 + (-0.0270 + 0.999i)T \)
31 \( 1 + (0.241 - 0.970i)T \)
37 \( 1 + (0.996 + 0.0811i)T \)
41 \( 1 + (0.395 - 0.918i)T \)
43 \( 1 + (0.108 + 0.994i)T \)
47 \( 1 + (0.161 - 0.986i)T \)
53 \( 1 + (0.827 - 0.561i)T \)
61 \( 1 + (0.0270 + 0.999i)T \)
67 \( 1 + (0.647 - 0.762i)T \)
71 \( 1 + (0.583 + 0.812i)T \)
73 \( 1 + (0.492 + 0.870i)T \)
79 \( 1 + (0.842 + 0.538i)T \)
83 \( 1 + (0.605 + 0.796i)T \)
89 \( 1 + (-0.725 + 0.687i)T \)
97 \( 1 + (-0.870 - 0.492i)T \)
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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

−18.82195935573322877556538054968, −18.229302611087438081568944491287, −17.227417304965449071992017920772, −16.33692902898145531407917435351, −15.74408279466342885127895983969, −15.24812516823861423742531241772, −14.69946000727786686669565496877, −13.99903296378269271530565358825, −13.37399345298041555487397231279, −12.46254754773976918160143226474, −11.47251151374764660422834531666, −11.201103161705050428143907460301, −10.49240926301985089766377372072, −9.315666165146399470362275267430, −8.95429631361333951353834042868, −8.29688528508886004658333656759, −7.68349184131832460931956565004, −6.68755042869123504870888063206, −6.063143941331778711825822862952, −4.967760562833320941886367353957, −4.25752385344586202680694605259, −3.52600832824774855913756272337, −2.89158535732745528579408894808, −2.18265174797739325424889415212, −1.02569654002015955048550685145, 0.82498008473496740088804468273, 1.283775277060045564799970622547, 2.271405338358955926016554666689, 3.41490887759124520426624932542, 3.97719252103491718974927004060, 4.448708091510959588510241047690, 5.5741122500342035348790735796, 6.72340969155581426268523771940, 7.11461530255164571832226585246, 8.024044436575431534202498606595, 8.31974288000223625916629330021, 9.15889340469532310762354418582, 9.88253543627736816066109800888, 10.82187270939763725005217392901, 11.463137681879644496912851313225, 12.41405773026264559967186763658, 12.80083429343875018399689932500, 13.48031357389568286998336052593, 14.2518071661668004778016858811, 14.95438254075297181516011096667, 15.32188561681319247798341340455, 16.45858980779842791056493221389, 16.80467592033528754263200352070, 17.74413046361491344453483726244, 18.35990503767715286010314652119

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