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Properties

Label 1-4000-4000.1379-r1-0-0
Degree $1$
Conductor $4000$
Sign $-0.444 - 0.895i$
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 + (0.951 − 0.309i)7-s + (−0.982 − 0.187i)9-s + (0.790 − 0.612i)11-s + (−0.827 + 0.562i)13-s + (0.968 − 0.248i)17-s + (−0.995 + 0.0941i)19-s + (−0.218 − 0.975i)21-s + (−0.684 + 0.728i)23-s + (−0.278 + 0.960i)27-s + (−0.661 + 0.750i)29-s + (−0.968 + 0.248i)31-s + (−0.535 − 0.844i)33-s + (0.960 − 0.278i)37-s + (0.481 + 0.876i)39-s + ⋯
L(s)  = 1  + (0.0941 − 0.995i)3-s + (0.951 − 0.309i)7-s + (−0.982 − 0.187i)9-s + (0.790 − 0.612i)11-s + (−0.827 + 0.562i)13-s + (0.968 − 0.248i)17-s + (−0.995 + 0.0941i)19-s + (−0.218 − 0.975i)21-s + (−0.684 + 0.728i)23-s + (−0.278 + 0.960i)27-s + (−0.661 + 0.750i)29-s + (−0.968 + 0.248i)31-s + (−0.535 − 0.844i)33-s + (0.960 − 0.278i)37-s + (0.481 + 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ -0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.124362652 - 1.812351712i\)
\(L(\frac12)\) \(\approx\) \(1.124362652 - 1.812351712i\)
\(L(1)\) \(\approx\) \(1.061106472 - 0.4339352379i\)
\(L(1)\) \(\approx\) \(1.061106472 - 0.4339352379i\)

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 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.790 - 0.612i)T \)
13 \( 1 + (-0.827 + 0.562i)T \)
17 \( 1 + (0.968 - 0.248i)T \)
19 \( 1 + (-0.995 + 0.0941i)T \)
23 \( 1 + (-0.684 + 0.728i)T \)
29 \( 1 + (-0.661 + 0.750i)T \)
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 \( 1 + (0.929 - 0.368i)T \)
53 \( 1 + (0.218 + 0.975i)T \)
59 \( 1 + (-0.940 - 0.338i)T \)
61 \( 1 + (0.0314 + 0.999i)T \)
67 \( 1 + (0.750 - 0.661i)T \)
71 \( 1 + (-0.368 - 0.929i)T \)
73 \( 1 + (0.904 + 0.425i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.995 - 0.0941i)T \)
89 \( 1 + (0.904 + 0.425i)T \)
97 \( 1 + (-0.0627 - 0.998i)T \)
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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

−18.55876963152116163863612342017, −17.58491108302912889419953059560, −17.11186959780836995378687829098, −16.68150405516180013152957517582, −15.638405717581920006594962190976, −15.03980061617688823882488246505, −14.52378860876854637093587927556, −14.22679240718293922048813067121, −12.96365261705616645154744419416, −12.19797082080618553630045959193, −11.68238384567073390046807099661, −10.80723249782583215982994240352, −10.292066309948461978954874709967, −9.50604833493984625580794472942, −8.949252416257213125885479881479, −8.04690123841621345051982126375, −7.616604591320465786855773349433, −6.43186817608533633548345452984, −5.59810551450771938162957434210, −5.04183350506229323359173339037, −4.1635770336280163184498894453, −3.789687184793566189092895007174, −2.471767078443486003198045325706, −2.05486613720521293057172641509, −0.72955011016468499598258021412, 0.38060135978688844072458428601, 1.36160253445687472633163176790, 1.82305040167467623368995840597, 2.79540833070096532165723494935, 3.734938177761187442840668147418, 4.52514027586203283057373780588, 5.52381417070160030238486414173, 6.086447688760025989708880266724, 7.0153079551499817627588217164, 7.5905782539802553449416994208, 8.13360231049845990778003842765, 9.00991196737212550638442938673, 9.57285801643093049773050354827, 10.80026284995459200351262636101, 11.26046531554137397349927694519, 12.03014392476957100107608262247, 12.46182895384366487273254764904, 13.41192093316353068114117719528, 14.0747917197704960899937729696, 14.5654623506557738345496958545, 14.97665486990067316887034471009, 16.49657764710214920233241530266, 16.72096341268684611686928888210, 17.457479940740954425345542673496, 18.16024517706656142774690073963

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