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Properties

Label 1-4000-4000.1197-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.989 - 0.145i$
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  + (0.397 − 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

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

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.989 - 0.145i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.989 - 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.577967662 - 0.1886355742i\)
\(L(\frac12)\) \(\approx\) \(2.577967662 - 0.1886355742i\)
\(L(1)\) \(\approx\) \(1.219660268 - 0.2075959505i\)
\(L(1)\) \(\approx\) \(1.219660268 - 0.2075959505i\)

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

−18.50802559033410689898495756751, −17.29898775694212392374740104436, −16.950382011074884982570673157607, −16.28109214834509589416922923347, −15.68861526861200238296264034100, −14.92216686921384281535924456790, −14.12347415837111563837668752114, −13.75465827565600128759087241503, −13.19874835690676705117665908096, −11.8144255946008564001249846996, −11.4153980834861232848151339809, −10.68310462978427526390951140862, −9.93126534614635768682160183622, −9.54400172157925717956899928005, −8.38099007216424668298332235634, −8.15042240332691053363846573901, −7.221566066331227875930848448806, −6.24026011001740903322396905880, −5.54110475006223915888464873987, −4.503976801130313604219066849589, −4.181879223368197856294577479589, −3.29212917234653801275831584638, −2.59045438886807862005690156883, −1.47023232127878590082756563325, −0.50386549959242725396012207965, 0.63786913716923762953948627210, 1.5973325764655214469831109913, 2.25967627921854893975039007948, 3.01043130450192159235352645077, 3.76227352706353629368038171495, 5.12310003106833781090649345359, 5.48568903446694323481947542932, 6.36328080233781835822740818764, 7.20598920380798315066132394987, 7.90815980838478266992896720859, 8.31067305667788404065829692629, 9.19091451127749907583533656494, 9.94037251848486711672714187969, 10.64418448601517114225371347498, 11.82968896265051476959147160199, 12.243981388622740271885728914074, 12.58666429069898374555605980952, 13.50923296644602617390407159207, 14.38653543324700924661694916154, 14.62704621626301238809155304900, 15.5873947810306745031675861108, 16.06791127394070545052734653434, 17.22841347373127631010235986494, 17.825079843764820082887986038105, 18.37077152358141473888637411411

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