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Properties

Label 1-10e3-1000.381-r0-0-0
Degree $1$
Conductor $1000$
Sign $0.492 - 0.870i$
Analytic cond. $4.64398$
Root an. cond. $4.64398$
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  + (0.929 + 0.368i)3-s + (0.309 − 0.951i)7-s + (0.728 + 0.684i)9-s + (−0.876 − 0.481i)11-s + (−0.728 − 0.684i)13-s + (0.535 − 0.844i)17-s + (0.929 − 0.368i)19-s + (0.637 − 0.770i)21-s + (−0.992 + 0.125i)23-s + (0.425 + 0.904i)27-s + (−0.968 + 0.248i)29-s + (0.535 − 0.844i)31-s + (−0.637 − 0.770i)33-s + (0.425 − 0.904i)37-s + (−0.425 − 0.904i)39-s + ⋯
L(s)  = 1  + (0.929 + 0.368i)3-s + (0.309 − 0.951i)7-s + (0.728 + 0.684i)9-s + (−0.876 − 0.481i)11-s + (−0.728 − 0.684i)13-s + (0.535 − 0.844i)17-s + (0.929 − 0.368i)19-s + (0.637 − 0.770i)21-s + (−0.992 + 0.125i)23-s + (0.425 + 0.904i)27-s + (−0.968 + 0.248i)29-s + (0.535 − 0.844i)31-s + (−0.637 − 0.770i)33-s + (0.425 − 0.904i)37-s + (−0.425 − 0.904i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(4.64398\)
Root analytic conductor: \(4.64398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1000,\ (0:\ ),\ 0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.627158636 - 0.9485503294i\)
\(L(\frac12)\) \(\approx\) \(1.627158636 - 0.9485503294i\)
\(L(1)\) \(\approx\) \(1.365744994 - 0.2126684857i\)
\(L(1)\) \(\approx\) \(1.365744994 - 0.2126684857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.929 + 0.368i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.876 - 0.481i)T \)
13 \( 1 + (-0.728 - 0.684i)T \)
17 \( 1 + (0.535 - 0.844i)T \)
19 \( 1 + (0.929 - 0.368i)T \)
23 \( 1 + (-0.992 + 0.125i)T \)
29 \( 1 + (-0.968 + 0.248i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (0.425 - 0.904i)T \)
41 \( 1 + (-0.992 - 0.125i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (0.0627 - 0.998i)T \)
53 \( 1 + (0.637 - 0.770i)T \)
59 \( 1 + (0.187 + 0.982i)T \)
61 \( 1 + (0.992 - 0.125i)T \)
67 \( 1 + (-0.968 - 0.248i)T \)
71 \( 1 + (0.0627 - 0.998i)T \)
73 \( 1 + (-0.187 + 0.982i)T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (0.929 - 0.368i)T \)
89 \( 1 + (-0.187 + 0.982i)T \)
97 \( 1 + (0.968 - 0.248i)T \)
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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

−21.67902411116616599496357699478, −20.89519623416321138552495496729, −20.31844209430406012704807060917, −19.3294566271235917243203418678, −18.705637029921313710008440945928, −18.13886715331512841259570285389, −17.232882841887103349120902752520, −16.04115458041082434686869102881, −15.36721530878829384911181920064, −14.623064941050395298235779358463, −14.02373205360309718849903717067, −13.00429540866013658278984448752, −12.274496433725375637053637561481, −11.70707705356118404522639212899, −10.2455252722185914773869097563, −9.66275664583466612324118865584, −8.731182992542668981918075167625, −7.9545208358961038613673442875, −7.35440436951866580598289789765, −6.20258180545140877917696716756, −5.247643878348286166309482263, −4.23157256764699508851128866194, −3.107575990984630238467231480889, −2.26315635474437840253281345917, −1.529479310136035170203286176925, 0.69647777797303111843404680288, 2.14170743433918059927082918687, 3.03491974605182855429892001880, 3.84672603423977393599794109901, 4.87439756172974240082740127490, 5.60799237526358747674395152313, 7.29692650531977407466631528047, 7.59154512596397256661065471416, 8.413404605286830953290789207354, 9.6008705212266953044427361926, 10.08551238828430872975655804188, 10.915497198039126715299077004848, 11.90730673605403139980513859220, 13.15866663631308675709151701367, 13.5893273285959107621830371447, 14.38930828728152034507321392640, 15.11643855760573142379129526180, 16.0650616685356493494602490809, 16.56107766550036276319996918194, 17.7323587860071778333130631598, 18.42252188749181038686394559658, 19.36100819859515313380831026013, 20.18877509411739212838460594712, 20.54483142886392905762457130849, 21.359500326185996625434351397987

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