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Properties

Label 1-4000-4000.237-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: $\chi_{4000} (237, \cdot )$
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

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

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