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Properties

Label 1-4000-4000.1467-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.145 - 0.989i$
Analytic cond. $18.5759$
Root an. cond. $18.5759$
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  + (0.397 + 0.917i)3-s + (0.309 − 0.951i)7-s + (−0.684 + 0.728i)9-s + (0.278 + 0.960i)11-s + (−0.0314 − 0.999i)13-s + (−0.844 − 0.535i)17-s + (−0.397 + 0.917i)19-s + (0.995 − 0.0941i)21-s + (−0.992 + 0.125i)23-s + (−0.940 − 0.338i)27-s + (0.509 − 0.860i)29-s + (−0.535 + 0.844i)31-s + (−0.770 + 0.637i)33-s + (−0.338 − 0.940i)37-s + (0.904 − 0.425i)39-s + ⋯
L(s)  = 1  + (0.397 + 0.917i)3-s + (0.309 − 0.951i)7-s + (−0.684 + 0.728i)9-s + (0.278 + 0.960i)11-s + (−0.0314 − 0.999i)13-s + (−0.844 − 0.535i)17-s + (−0.397 + 0.917i)19-s + (0.995 − 0.0941i)21-s + (−0.992 + 0.125i)23-s + (−0.940 − 0.338i)27-s + (0.509 − 0.860i)29-s + (−0.535 + 0.844i)31-s + (−0.770 + 0.637i)33-s + (−0.338 − 0.940i)37-s + (0.904 − 0.425i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.145 - 0.989i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.145 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6777688707 - 0.5853441107i\)
\(L(\frac12)\) \(\approx\) \(0.6777688707 - 0.5853441107i\)
\(L(1)\) \(\approx\) \(1.001321041 + 0.1356497494i\)
\(L(1)\) \(\approx\) \(1.001321041 + 0.1356497494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.397 + 0.917i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.278 + 0.960i)T \)
13 \( 1 + (-0.0314 - 0.999i)T \)
17 \( 1 + (-0.844 - 0.535i)T \)
19 \( 1 + (-0.397 + 0.917i)T \)
23 \( 1 + (-0.992 + 0.125i)T \)
29 \( 1 + (0.509 - 0.860i)T \)
31 \( 1 + (-0.535 + 0.844i)T \)
37 \( 1 + (-0.338 - 0.940i)T \)
41 \( 1 + (-0.125 + 0.992i)T \)
43 \( 1 + (-0.987 + 0.156i)T \)
47 \( 1 + (0.998 + 0.0627i)T \)
53 \( 1 + (-0.0941 - 0.995i)T \)
59 \( 1 + (0.562 - 0.827i)T \)
61 \( 1 + (0.612 - 0.790i)T \)
67 \( 1 + (0.509 + 0.860i)T \)
71 \( 1 + (-0.998 - 0.0627i)T \)
73 \( 1 + (0.187 - 0.982i)T \)
79 \( 1 + (0.929 + 0.368i)T \)
83 \( 1 + (-0.917 - 0.397i)T \)
89 \( 1 + (0.982 + 0.187i)T \)
97 \( 1 + (0.248 + 0.968i)T \)
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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

−18.68656552219309603137633778586, −18.13300606035491096930756284123, −17.37243067441186098837070209932, −16.745587128985636196865424886554, −15.802573087051455044721046330355, −15.16420620248286491652902830041, −14.45086454403424543530282706113, −13.81544766198478949646618215652, −13.27101532048696632929466533833, −12.44577154072836628126109343130, −11.76618956504779593320987384048, −11.36914575976702523739357487229, −10.460325451967954609884815749117, −9.235897137144241591227490319372, −8.756306380345333159566753664754, −8.43062311345482265777498262883, −7.42049226618184031430525908771, −6.601689188678203541555782600984, −6.157004318779398946791319455383, −5.34259849857260495443874874880, −4.30799713449258367216044832680, −3.48621967500880212693247492087, −2.464085772555156379671854765517, −2.05597228287883451661642951023, −1.11195511960028712564086675663, 0.22966252383294898344678138077, 1.651775845227844833384256354945, 2.385923530328868052726488635427, 3.45397893016121222894906540325, 4.02181317986934762233282271459, 4.69248916902007828732951680073, 5.35223188155390364055116954006, 6.36800647468122892277278231256, 7.236666552607137762726062089705, 7.98567407425771326025037174739, 8.48760936855210461776123930429, 9.54603030711681629168194699218, 10.03581545678629795168322707430, 10.55772571865488088908839934693, 11.2841829950776634695811392086, 12.10311878019289068162926013466, 12.98591979422132764343889340171, 13.673421490140212242718250170148, 14.42388637211089406826413140717, 14.81471436980320923089102082015, 15.682682585238186178873139336344, 16.1548105693611499967994246882, 16.970161029732984156448636526325, 17.643256662082974184948760494620, 18.03606163728479337151185869103

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