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Properties

Label 1-4003-4003.3552-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.940 - 0.338i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (0.460 − 0.887i)2-s + (−0.576 − 0.816i)3-s + (−0.576 − 0.816i)4-s + (0.460 + 0.887i)5-s + (−0.990 + 0.136i)6-s + (0.682 − 0.730i)7-s + (−0.990 + 0.136i)8-s + (−0.334 + 0.942i)9-s + 10-s + (0.203 − 0.979i)11-s + (−0.334 + 0.942i)12-s + (−0.334 − 0.942i)13-s + (−0.334 − 0.942i)14-s + (0.460 − 0.887i)15-s + (−0.334 + 0.942i)16-s + (−0.990 − 0.136i)17-s + ⋯
L(s)  = 1  + (0.460 − 0.887i)2-s + (−0.576 − 0.816i)3-s + (−0.576 − 0.816i)4-s + (0.460 + 0.887i)5-s + (−0.990 + 0.136i)6-s + (0.682 − 0.730i)7-s + (−0.990 + 0.136i)8-s + (−0.334 + 0.942i)9-s + 10-s + (0.203 − 0.979i)11-s + (−0.334 + 0.942i)12-s + (−0.334 − 0.942i)13-s + (−0.334 − 0.942i)14-s + (0.460 − 0.887i)15-s + (−0.334 + 0.942i)16-s + (−0.990 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.940 - 0.338i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (3552, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.940 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.124802345 - 0.1963504631i\)
\(L(\frac12)\) \(\approx\) \(1.124802345 - 0.1963504631i\)
\(L(1)\) \(\approx\) \(0.8390073197 - 0.5893962310i\)
\(L(1)\) \(\approx\) \(0.8390073197 - 0.5893962310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.460 - 0.887i)T \)
3 \( 1 + (-0.576 - 0.816i)T \)
5 \( 1 + (0.460 + 0.887i)T \)
7 \( 1 + (0.682 - 0.730i)T \)
11 \( 1 + (0.203 - 0.979i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (-0.990 - 0.136i)T \)
19 \( 1 + (0.460 + 0.887i)T \)
23 \( 1 + (-0.775 + 0.631i)T \)
29 \( 1 + (0.682 + 0.730i)T \)
31 \( 1 + (-0.0682 + 0.997i)T \)
37 \( 1 + (-0.0682 + 0.997i)T \)
41 \( 1 + (-0.0682 + 0.997i)T \)
43 \( 1 + (-0.0682 - 0.997i)T \)
47 \( 1 + (0.203 - 0.979i)T \)
53 \( 1 + (-0.990 - 0.136i)T \)
59 \( 1 + (0.682 + 0.730i)T \)
61 \( 1 + (0.682 + 0.730i)T \)
67 \( 1 + (0.962 + 0.269i)T \)
71 \( 1 + (-0.775 - 0.631i)T \)
73 \( 1 + (-0.576 + 0.816i)T \)
79 \( 1 + (-0.576 + 0.816i)T \)
83 \( 1 + (0.203 + 0.979i)T \)
89 \( 1 + (-0.576 + 0.816i)T \)
97 \( 1 + (-0.0682 + 0.997i)T \)
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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

−17.91520011266213628849891512740, −17.60724773352606962752498866509, −17.2419195232266628788848621708, −16.320134032859722605457652735005, −15.79941096986908348035420102456, −15.32468380978189594619761951474, −14.4577074352351749647794417726, −14.05583406426238424059754661213, −12.97057844827178711818472396644, −12.406488590370737911197089305804, −11.761974307964071066044992618134, −11.21225634757304733907566232050, −9.891313688205000488965669702564, −9.35652519734412921190624865138, −8.88859186005420096909616513859, −8.14230528700599168543547151750, −7.13357000450607537904017038948, −6.294497459204568600812676053646, −5.77978482815927770368540233484, −4.86188477722098878192003509169, −4.546253345716981368569750311937, −4.09465239253606211866319100638, −2.61436554029650988199573787960, −1.870142448851302237294654752741, −0.31765043032127001364007347592, 1.01016599852299330261657884601, 1.57025727487756812591632454072, 2.495009608787614695776616267372, 3.20968976047556960319663589522, 4.01894831600914000534252294145, 5.1942047186613443280030266130, 5.51486696979309195781394667932, 6.450562577789265862161465202378, 6.98996053540565798411266072192, 8.00232400391472602469636022095, 8.59793403434656731361298909532, 9.911003905413768685734762470475, 10.412976501288802110381612519017, 10.9443656763099925286325386249, 11.57470644614468232887240101863, 12.10050490308498462550758529925, 13.11523325366189387327458072710, 13.603552041300819968124150810, 14.096143766332389646621001068631, 14.63482705279349146895078310755, 15.61319506971340574449258646481, 16.58754237862511708096828965976, 17.552201471546707685056246972704, 17.8388203276766993027792151403, 18.40650801227174414326848968698

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