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Properties

Label 1-4000-4000.53-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.244 - 0.969i$
Analytic cond. $429.859$
Root an. cond. $429.859$
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  + (−0.917 − 0.397i)3-s + (0.309 + 0.951i)7-s + (0.684 + 0.728i)9-s + (0.960 + 0.278i)11-s + (−0.999 − 0.0314i)13-s + (−0.844 + 0.535i)17-s + (−0.917 + 0.397i)19-s + (0.0941 − 0.995i)21-s + (−0.992 − 0.125i)23-s + (−0.338 − 0.940i)27-s + (−0.860 + 0.509i)29-s + (0.535 + 0.844i)31-s + (−0.770 − 0.637i)33-s + (0.940 + 0.338i)37-s + (0.904 + 0.425i)39-s + ⋯
L(s)  = 1  + (−0.917 − 0.397i)3-s + (0.309 + 0.951i)7-s + (0.684 + 0.728i)9-s + (0.960 + 0.278i)11-s + (−0.999 − 0.0314i)13-s + (−0.844 + 0.535i)17-s + (−0.917 + 0.397i)19-s + (0.0941 − 0.995i)21-s + (−0.992 − 0.125i)23-s + (−0.338 − 0.940i)27-s + (−0.860 + 0.509i)29-s + (0.535 + 0.844i)31-s + (−0.770 − 0.637i)33-s + (0.940 + 0.338i)37-s + (0.904 + 0.425i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1076087911 - 0.08387678004i\)
\(L(\frac12)\) \(\approx\) \(0.1076087911 - 0.08387678004i\)
\(L(1)\) \(\approx\) \(0.6494102129 + 0.1086764010i\)
\(L(1)\) \(\approx\) \(0.6494102129 + 0.1086764010i\)

Euler product

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

−18.273507051532791978758036085550, −17.39609300817433596754447008950, −17.27752051920654574788660355029, −16.53319852370642557172629229267, −15.92963921781603380444557017495, −14.97026854985912501052489692446, −14.55688511220138410008129088316, −13.608063668965735764411365661078, −12.99568534085435343452610289052, −12.09010294508148998376285850504, −11.44301052398563275675679320037, −11.05092325214318581844093721818, −10.16092310721596637675399769878, −9.61999215771009508788573859950, −8.91033283788232615936741947509, −7.790934478858882656800636489796, −7.15740421066087634396296493438, −6.42120075158161531619088012450, −5.84789456708414937182073050152, −4.70966869866891527493207904729, −4.35365056338507410773672096395, −3.721753345482192164032553526964, −2.448445027644075851608575551508, −1.51056588249235686827369095350, −0.47426768174655441181055871390, 0.04096357306475288399748782620, 1.47079765984035267218375743085, 1.91366292154999737346867847984, 2.81966231186433014139509164709, 4.21619437164733121518123054050, 4.589783276217420619940469299759, 5.56363644390567425569659679718, 6.17997462686783323310128004910, 6.748519210774080561141211341284, 7.59963044742097802977706970231, 8.39009715558797629433094275806, 9.12669620508014367811807014393, 9.96208351289295145394754445833, 10.66351962881371815106696104094, 11.469896111567297595442843888133, 12.048545989081841618330186289703, 12.46643753831745239367686299129, 13.17356749368217575349935363284, 14.1333817096662622051601978639, 14.95512095143769367881715852432, 15.27556256835984392599650714772, 16.4291116495888757995581186887, 16.78912422505514755432711220773, 17.62120118032500013616069456509, 18.01572763400904197338735446445

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