↑

Properties

Label 1-4001-4001.145-r0-0-0
Degree $1$
Conductor $4001$
Sign $0.326 + 0.945i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.567 + 0.823i)2-s + (0.585 − 0.810i)3-s + (−0.356 + 0.934i)4-s + (−0.844 + 0.535i)5-s + (0.999 + 0.0219i)6-s + (−0.391 − 0.920i)7-s + (−0.971 + 0.236i)8-s + (−0.314 − 0.949i)9-s + (−0.920 − 0.391i)10-s + (−0.972 − 0.233i)11-s + (0.549 + 0.835i)12-s + (−0.414 + 0.910i)13-s + (0.535 − 0.844i)14-s + (−0.0596 + 0.998i)15-s + (−0.745 − 0.666i)16-s + (−0.663 − 0.748i)17-s + ⋯
L(s)  = 1  + (0.567 + 0.823i)2-s + (0.585 − 0.810i)3-s + (−0.356 + 0.934i)4-s + (−0.844 + 0.535i)5-s + (0.999 + 0.0219i)6-s + (−0.391 − 0.920i)7-s + (−0.971 + 0.236i)8-s + (−0.314 − 0.949i)9-s + (−0.920 − 0.391i)10-s + (−0.972 − 0.233i)11-s + (0.549 + 0.835i)12-s + (−0.414 + 0.910i)13-s + (0.535 − 0.844i)14-s + (−0.0596 + 0.998i)15-s + (−0.745 − 0.666i)16-s + (−0.663 − 0.748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8924413688 + 0.6357292363i\)
\(L(\frac12)\) \(\approx\) \(0.8924413688 + 0.6357292363i\)
\(L(1)\) \(\approx\) \(0.9898903526 + 0.2385111660i\)
\(L(1)\) \(\approx\) \(0.9898903526 + 0.2385111660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (0.567 + 0.823i)T \)
3 \( 1 + (0.585 - 0.810i)T \)
5 \( 1 + (-0.844 + 0.535i)T \)
7 \( 1 + (-0.391 - 0.920i)T \)
11 \( 1 + (-0.972 - 0.233i)T \)
13 \( 1 + (-0.414 + 0.910i)T \)
17 \( 1 + (-0.663 - 0.748i)T \)
19 \( 1 + (-0.925 + 0.379i)T \)
23 \( 1 + (0.341 - 0.939i)T \)
29 \( 1 + (0.362 + 0.932i)T \)
31 \( 1 + (-0.362 + 0.932i)T \)
37 \( 1 + (-0.944 - 0.329i)T \)
41 \( 1 + (0.776 + 0.630i)T \)
43 \( 1 + (0.999 - 0.0282i)T \)
47 \( 1 + (0.967 - 0.254i)T \)
53 \( 1 + (0.323 - 0.946i)T \)
59 \( 1 + (-0.827 - 0.562i)T \)
61 \( 1 + (-0.181 + 0.983i)T \)
67 \( 1 + (0.686 - 0.726i)T \)
71 \( 1 + (-0.470 - 0.882i)T \)
73 \( 1 + (0.780 - 0.625i)T \)
79 \( 1 + (-0.998 - 0.0627i)T \)
83 \( 1 + (0.487 + 0.873i)T \)
89 \( 1 + (0.314 + 0.949i)T \)
97 \( 1 + (0.291 + 0.956i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.9112466260913938740083106857, −17.67833650298303555417378838008, −17.009446448686670685354521448290, −15.68127388752943281958586983679, −15.4071167897913931691176919219, −15.337623827003220198013555914615, −14.312487792382584874461729584296, −13.29440016809431376207617833781, −12.865439620599065780526903925409, −12.33995426086894440729532978024, −11.39614938661584909593702326769, −10.805843928503409556048919726181, −10.16117547225516463266478963693, −9.36072272162714800175369649588, −8.78259475862236290591088622092, −8.15622067086317687454665669661, −7.2961273006172146604840731809, −5.82750787229721777656852672356, −5.477039046645252929227982039000, −4.53246496906047368489116136693, −4.13023045048852940584947629627, −3.17333778439536911306347155146, −2.60507644176217512978886901874, −1.932394466301043305341243635984, −0.361762537718822562252949871053, 0.60525557726125387877203905998, 2.226786788766112052892777687338, 2.87844399573954034273153297930, 3.62174983638895470273899179881, 4.30577020929014937303574059833, 5.06241131493373767844986975807, 6.37770128674190011513360985621, 6.75256773975424469826605843175, 7.320453171209655079273745975221, 7.85583622255274531174193651638, 8.64960952435099250539766293845, 9.22995877603910169954228739340, 10.517361139404310555237731603908, 11.08220243107396956475592484572, 12.19386249682297615706197801979, 12.51859015823030419276974329557, 13.31078089182721486437779736068, 13.985590287162193856034542289802, 14.41669561374845159671224935016, 15.060084110127895728345481679757, 15.96486635866831238140855519248, 16.316198134729160324396098821445, 17.19554538546606167402103150980, 18.01613974402251675956595705785, 18.54356090977870022475353774941

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