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Properties

Label 1-4000-4000.2717-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: $\chi_{4000} (2717, \cdot )$
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

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

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