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Properties

Label 1-4000-4000.1517-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.0674 + 0.997i$
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  + (−0.0941 − 0.995i)3-s + (0.309 − 0.951i)7-s + (−0.982 + 0.187i)9-s + (0.612 − 0.790i)11-s + (−0.827 − 0.562i)13-s + (0.248 − 0.968i)17-s + (−0.0941 + 0.995i)19-s + (−0.975 − 0.218i)21-s + (0.728 − 0.684i)23-s + (0.278 + 0.960i)27-s + (0.750 − 0.661i)29-s + (0.968 + 0.248i)31-s + (−0.844 − 0.535i)33-s + (0.960 + 0.278i)37-s + (−0.481 + 0.876i)39-s + ⋯
L(s)  = 1  + (−0.0941 − 0.995i)3-s + (0.309 − 0.951i)7-s + (−0.982 + 0.187i)9-s + (0.612 − 0.790i)11-s + (−0.827 − 0.562i)13-s + (0.248 − 0.968i)17-s + (−0.0941 + 0.995i)19-s + (−0.975 − 0.218i)21-s + (0.728 − 0.684i)23-s + (0.278 + 0.960i)27-s + (0.750 − 0.661i)29-s + (0.968 + 0.248i)31-s + (−0.844 − 0.535i)33-s + (0.960 + 0.278i)37-s + (−0.481 + 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0674 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0674 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.0674 + 0.997i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.0674 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4641416521 - 0.4966002864i\)
\(L(\frac12)\) \(\approx\) \(-0.4641416521 - 0.4966002864i\)
\(L(1)\) \(\approx\) \(0.7893296462 - 0.5454361767i\)
\(L(1)\) \(\approx\) \(0.7893296462 - 0.5454361767i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.0941 - 0.995i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.612 - 0.790i)T \)
13 \( 1 + (-0.827 - 0.562i)T \)
17 \( 1 + (0.248 - 0.968i)T \)
19 \( 1 + (-0.0941 + 0.995i)T \)
23 \( 1 + (0.728 - 0.684i)T \)
29 \( 1 + (0.750 - 0.661i)T \)
31 \( 1 + (0.968 + 0.248i)T \)
37 \( 1 + (0.960 + 0.278i)T \)
41 \( 1 + (-0.684 + 0.728i)T \)
43 \( 1 + (-0.156 - 0.987i)T \)
47 \( 1 + (-0.368 + 0.929i)T \)
53 \( 1 + (-0.218 + 0.975i)T \)
59 \( 1 + (-0.338 - 0.940i)T \)
61 \( 1 + (-0.999 - 0.0314i)T \)
67 \( 1 + (0.750 + 0.661i)T \)
71 \( 1 + (-0.368 + 0.929i)T \)
73 \( 1 + (-0.425 - 0.904i)T \)
79 \( 1 + (-0.637 + 0.770i)T \)
83 \( 1 + (-0.995 - 0.0941i)T \)
89 \( 1 + (0.904 - 0.425i)T \)
97 \( 1 + (0.998 + 0.0627i)T \)
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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

−18.91598201984156359309485526019, −17.906180641482397219485626905774, −17.3508878972976542251428477123, −16.88750570773522382685479308671, −15.99727182736946264941505223829, −15.28338705478963518039165143443, −14.8600488068439125885018600769, −14.4113979038970189115938873600, −13.37856305766130770442601296604, −12.45290701542703032200871443887, −11.82997221096078917693269300468, −11.36269656721542599877821104863, −10.45287290699466935063258266265, −9.74960751828251598986657660925, −9.1594371578952229029771907211, −8.64502457159968134776155184112, −7.74003880903600004522524701335, −6.735401366895865860320403378240, −6.105580700853409426004663937407, −5.050293256083920237643591485104, −4.78495515521273653473942779297, −3.92441950708148580091650616200, −2.97196716464309942895652922991, −2.27489642993904015935966228790, −1.31014963218605411095120189435, 0.1141932503302206187783956102, 0.88618523421058937295767334282, 1.423824951641385378071970567065, 2.678441820887088445037031188644, 3.14829333775194343203137941902, 4.3135406187646038786283395381, 5.02255020895570486737499350896, 5.959170382436862642674347241971, 6.60802908431722424296880649567, 7.28404224361404591210644662025, 7.99508721414196556966015010055, 8.44238527178848380053352009007, 9.52713843564276771126526851479, 10.264398971941206385840198458791, 11.035521794761478232263130783860, 11.733659274986436039385925133332, 12.26547725060889148240175151584, 13.08292512087587033911407361403, 13.765647696269147005617211340618, 14.25026823035755888692195781904, 14.79360862234996508845783159130, 15.92333805567904437778738076261, 16.795617925802491305536503042595, 17.08519658587242571231387588889, 17.72763430236691448070213449310

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