Properties

Label 1-4000-4000.3173-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.891 + 0.452i$
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  + (0.612 + 0.790i)3-s + (−0.809 − 0.587i)7-s + (−0.248 + 0.968i)9-s + (0.995 − 0.0941i)11-s + (0.509 + 0.860i)13-s + (−0.982 − 0.187i)17-s + (0.612 − 0.790i)19-s + (−0.0314 − 0.999i)21-s + (0.535 + 0.844i)23-s + (−0.917 + 0.397i)27-s + (0.338 + 0.940i)29-s + (−0.187 + 0.982i)31-s + (0.684 + 0.728i)33-s + (−0.397 + 0.917i)37-s + (−0.368 + 0.929i)39-s + ⋯
L(s)  = 1  + (0.612 + 0.790i)3-s + (−0.809 − 0.587i)7-s + (−0.248 + 0.968i)9-s + (0.995 − 0.0941i)11-s + (0.509 + 0.860i)13-s + (−0.982 − 0.187i)17-s + (0.612 − 0.790i)19-s + (−0.0314 − 0.999i)21-s + (0.535 + 0.844i)23-s + (−0.917 + 0.397i)27-s + (0.338 + 0.940i)29-s + (−0.187 + 0.982i)31-s + (0.684 + 0.728i)33-s + (−0.397 + 0.917i)37-s + (−0.368 + 0.929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.891 + 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4932398316 + 2.061623582i\)
\(L(\frac12)\) \(\approx\) \(0.4932398316 + 2.061623582i\)
\(L(1)\) \(\approx\) \(1.138054658 + 0.4643726960i\)
\(L(1)\) \(\approx\) \(1.138054658 + 0.4643726960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.612 + 0.790i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.995 - 0.0941i)T \)
13 \( 1 + (0.509 + 0.860i)T \)
17 \( 1 + (-0.982 - 0.187i)T \)
19 \( 1 + (0.612 - 0.790i)T \)
23 \( 1 + (0.535 + 0.844i)T \)
29 \( 1 + (0.338 + 0.940i)T \)
31 \( 1 + (-0.187 + 0.982i)T \)
37 \( 1 + (-0.397 + 0.917i)T \)
41 \( 1 + (-0.844 - 0.535i)T \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (0.481 + 0.876i)T \)
53 \( 1 + (0.999 - 0.0314i)T \)
59 \( 1 + (0.661 - 0.750i)T \)
61 \( 1 + (-0.975 - 0.218i)T \)
67 \( 1 + (0.338 - 0.940i)T \)
71 \( 1 + (0.481 + 0.876i)T \)
73 \( 1 + (0.0627 - 0.998i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (-0.790 - 0.612i)T \)
89 \( 1 + (-0.998 - 0.0627i)T \)
97 \( 1 + (0.904 + 0.425i)T \)
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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

−18.36632344061723509543999703849, −17.38376743380097664280310207982, −16.79920307464630904031133727496, −15.808907283437171677634302206853, −15.244274089970574494684438908867, −14.65049684511876088331276277586, −13.80059257048029961757423605525, −13.2321570381002633420492922505, −12.61062803375135323368581380757, −12.01636911414456218902616960791, −11.34304795814231715316972652165, −10.270758421906645480113059868419, −9.58149760765179397857367028730, −8.763450547418345762700825457679, −8.48765983355823323484491272858, −7.43171027411390374530163252955, −6.8004370322423058968248048699, −6.07865014807972805202124280952, −5.59994766167554741247509605101, −4.15223025190392196783347969832, −3.61663047435806084310015543636, −2.7102238598544434548899494471, −2.12422089079182557337687252004, −1.072520535297860873327196459037, −0.31425580310676362286170459873, 1.00350394598944353458369988381, 1.88250196609733818324971555875, 3.0714812408887650425125087368, 3.44239572503473789643693837450, 4.309222597830320378590829547712, 4.84991037510983294689805158683, 5.92110469047885761840307482071, 6.939140564474534505143054215709, 7.10909695138633743317981422641, 8.44696643416095630921939556470, 9.085984926398051721356953984619, 9.3781624883950988234235457375, 10.24174130105432306199524222759, 11.027551920688906665698733639323, 11.489705946503919691474282326826, 12.519922252013941398293350115633, 13.43815585521405380509216335705, 13.86541131713434599280687570891, 14.359659591890702897843330332582, 15.44252303902389553454587734077, 15.77777183589332778083653230592, 16.49038706453275485520421647533, 17.07061712969906442789855960328, 17.802379895054151874786370510188, 18.863390928268321111225497294

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