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Properties

Label 1-4004-4004.159-r0-0-0
Degree $1$
Conductor $4004$
Sign $-0.999 + 0.00901i$
Analytic cond. $18.5944$
Root an. cond. $18.5944$
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  + (0.669 − 0.743i)3-s + (0.104 − 0.994i)5-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)15-s + (0.809 + 0.587i)17-s + (−0.978 − 0.207i)19-s − 23-s + (−0.978 − 0.207i)25-s + (−0.809 − 0.587i)27-s + (0.669 + 0.743i)29-s + (−0.104 − 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 + 0.207i)41-s + (0.5 + 0.866i)43-s − 45-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (0.104 − 0.994i)5-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)15-s + (0.809 + 0.587i)17-s + (−0.978 − 0.207i)19-s − 23-s + (−0.978 − 0.207i)25-s + (−0.809 − 0.587i)27-s + (0.669 + 0.743i)29-s + (−0.104 − 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 + 0.207i)41-s + (0.5 + 0.866i)43-s − 45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.999 + 0.00901i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ -0.999 + 0.00901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006882663132 - 1.526323899i\)
\(L(\frac12)\) \(\approx\) \(0.006882663132 - 1.526323899i\)
\(L(1)\) \(\approx\) \(1.007287575 - 0.6555848382i\)
\(L(1)\) \(\approx\) \(1.007287575 - 0.6555848382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 - T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−18.92268962773751907904777904623, −18.28300438959672773323587252121, −17.413741449878187759066434685439, −16.74913105128777707821625946768, −15.85853817918394705400231296964, −15.45948241647217317478166250458, −14.69040599203658559551886249314, −14.010450674423351195579231626433, −13.846807418897319334587138278169, −12.66429337097135101594252846642, −11.89823740214509022820658735626, −11.03259220246452514363881698484, −10.42618808315958215283900806947, −9.93685213492737721502486742043, −9.23473569695008348648641193091, −8.33903922258294528620260635968, −7.75316996069499454146872785489, −6.99757719732040384585119782524, −6.08821390697018962949715280797, −5.421884381204542357475289479670, −4.3092385824393758565080881355, −3.88454240919171943746016652709, −2.791104317521574978952249008094, −2.56813721701435846752261460341, −1.41029474484362486725601776380, 0.36472191086867523803122141683, 1.35084643325597373174153963897, 2.00644274154701286107997959914, 2.842300469852156845744335070380, 3.91547570317269013059398786742, 4.38290163843482677535065086660, 5.6100587632256477738570666757, 6.064942528715319727608358308185, 6.98905478388447538971541150113, 7.912966507035184747568912057569, 8.23580141371648989528270435877, 9.03065183954875231825532982435, 9.6137583352904829499040114085, 10.43904287369142606363322512058, 11.464030065198854012502703909225, 12.23579297661671035584263477614, 12.74198820604098837747357714621, 13.1846788451611258417038121455, 14.08335845344786070764691849863, 14.565311551126149439023668326940, 15.35458652393700922656448233411, 16.193774373822341163704823017249, 16.78287594411435699929497442872, 17.61618755260623120018692883231, 18.0305797414613143995846370394

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