↑

Properties

Label 1-4001-4001.3849-r0-0-0
Degree $1$
Conductor $4001$
Sign $-0.0956 + 0.995i$
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  + (−0.809 − 0.587i)2-s + (0.951 + 0.309i)3-s + (0.309 + 0.951i)4-s + 5-s + (−0.587 − 0.809i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)9-s + (−0.809 − 0.587i)10-s + i·11-s + i·12-s + (0.309 + 0.951i)13-s + 14-s + (0.951 + 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.951 + 0.309i)3-s + (0.309 + 0.951i)4-s + 5-s + (−0.587 − 0.809i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)9-s + (−0.809 − 0.587i)10-s + i·11-s + i·12-s + (0.309 + 0.951i)13-s + 14-s + (0.951 + 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $-0.0956 + 0.995i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (3849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ -0.0956 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191662040 + 1.311702173i\)
\(L(\frac12)\) \(\approx\) \(1.191662040 + 1.311702173i\)
\(L(1)\) \(\approx\) \(1.095510674 + 0.2457924173i\)
\(L(1)\) \(\approx\) \(1.095510674 + 0.2457924173i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + iT \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 - iT \)
59 \( 1 - T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.951 - 0.309i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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

−18.39243552740042981474811665754, −17.49699903614815595564324501454, −17.227936102282193401173268763147, −16.11928666429659152680778425538, −15.7209105827345137784081481533, −15.03615562249243074351629367139, −13.9781460820184115379248705090, −13.72881022630314766596439970166, −13.23379165361360923680146204651, −12.30866925075653225465216643595, −11.03403398856032029531671772690, −10.37657109632028778101805263783, −9.91998485174423692490321178176, −9.11013081543387639603292275919, −8.600218218946047763176317407696, −8.00045071411425774750613736194, −6.99438912138390880902872336081, −6.52170285278657424202826439139, −5.945815292517029455258287502588, −4.98778330121196180498846324998, −3.857090630837898944918714995670, −2.86253832156303726645810079556, −2.36527017475436422420763262118, −1.24395999452579533599664872060, −0.565214861697446589411138175072, 1.34663400525875872477818188745, 2.136608739816114943034323653567, 2.42479743697246080140399993336, 3.39639644404943619108143190048, 4.15573454849891173660338961726, 4.99861474715442594001665419934, 6.369818344234230498359697165065, 6.696462513244225115615142576646, 7.73049476667923588231777382533, 8.47038059888833757610834747822, 9.273711515542565041625007386122, 9.515348791484909628252612305509, 10.09910930700517021543306460617, 10.75177372750885337599366433618, 11.94591861011513439103709468230, 12.38543402008244719285701667876, 13.27677082945488594980433411233, 13.717899394600909930269735614853, 14.49997142457782337039473465754, 15.47385889704026436097592656539, 16.03385535160986837954773365904, 16.548941135554265696752622568889, 17.57552915567927287489265405242, 18.09572434716458405160078310850, 18.65887043932160167341797694903

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