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Properties

Label 1-4003-4003.1018-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.810 + 0.585i$
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  + (0.0415 − 0.999i)2-s + (0.0290 − 0.999i)3-s + (−0.996 − 0.0831i)4-s + (0.998 − 0.0596i)5-s + (−0.997 − 0.0705i)6-s + (−0.681 − 0.731i)7-s + (−0.124 + 0.992i)8-s + (−0.998 − 0.0580i)9-s + (−0.0180 − 0.999i)10-s + (−0.513 − 0.858i)11-s + (−0.112 + 0.993i)12-s + (0.356 + 0.934i)13-s + (−0.759 + 0.650i)14-s + (−0.0306 − 0.999i)15-s + (0.986 + 0.165i)16-s + (−0.989 − 0.145i)17-s + ⋯
L(s)  = 1  + (0.0415 − 0.999i)2-s + (0.0290 − 0.999i)3-s + (−0.996 − 0.0831i)4-s + (0.998 − 0.0596i)5-s + (−0.997 − 0.0705i)6-s + (−0.681 − 0.731i)7-s + (−0.124 + 0.992i)8-s + (−0.998 − 0.0580i)9-s + (−0.0180 − 0.999i)10-s + (−0.513 − 0.858i)11-s + (−0.112 + 0.993i)12-s + (0.356 + 0.934i)13-s + (−0.759 + 0.650i)14-s + (−0.0306 − 0.999i)15-s + (0.986 + 0.165i)16-s + (−0.989 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1018, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2284163510 - 0.07387709281i\)
\(L(\frac12)\) \(\approx\) \(-0.2284163510 - 0.07387709281i\)
\(L(1)\) \(\approx\) \(0.4798786222 - 0.6957585833i\)
\(L(1)\) \(\approx\) \(0.4798786222 - 0.6957585833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.0415 - 0.999i)T \)
3 \( 1 + (0.0290 - 0.999i)T \)
5 \( 1 + (0.998 - 0.0596i)T \)
7 \( 1 + (-0.681 - 0.731i)T \)
11 \( 1 + (-0.513 - 0.858i)T \)
13 \( 1 + (0.356 + 0.934i)T \)
17 \( 1 + (-0.989 - 0.145i)T \)
19 \( 1 + (0.930 + 0.365i)T \)
23 \( 1 + (0.362 - 0.931i)T \)
29 \( 1 + (-0.427 - 0.903i)T \)
31 \( 1 + (-0.453 + 0.891i)T \)
37 \( 1 + (0.605 - 0.795i)T \)
41 \( 1 + (-0.143 - 0.989i)T \)
43 \( 1 + (-0.914 + 0.404i)T \)
47 \( 1 + (0.0384 - 0.999i)T \)
53 \( 1 + (-0.838 + 0.544i)T \)
59 \( 1 + (-0.985 + 0.170i)T \)
61 \( 1 + (-0.667 - 0.744i)T \)
67 \( 1 + (-0.0932 + 0.995i)T \)
71 \( 1 + (-0.653 + 0.757i)T \)
73 \( 1 + (-0.757 - 0.652i)T \)
79 \( 1 + (-0.478 - 0.878i)T \)
83 \( 1 + (-0.177 + 0.984i)T \)
89 \( 1 + (-0.0431 + 0.999i)T \)
97 \( 1 + (0.178 + 0.983i)T \)
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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

−18.78332934404229940837734693238, −18.05895666370727211023273702661, −17.72200846221239276766272571077, −16.95731668204814580155545030176, −16.271119304541579476034343459673, −15.564866147277096437819691662550, −15.19752013781467015589052218663, −14.66685686213051934011884414851, −13.5996570095625421207337931054, −13.20377936031762450412590328102, −12.58408270103219777622809668085, −11.4163229239526234023574592773, −10.54061212256719932678597898688, −9.75498854081079608752958089944, −9.46164932941653838980038741320, −8.84953670994693728992113178964, −7.99411128925110657998742712638, −7.09527978189078449292242210579, −6.20680999379414783740129120308, −5.685688985883927348464538562449, −5.09064154265377501140938112231, −4.48571969097025736042446357622, −3.214832778870617018184517818502, −2.90391805858508907417037012329, −1.587071401493052998272264290060, 0.068774838285389880603734481502, 1.04494007951305226008345817897, 1.7577256322127240563719814801, 2.57201265811001152362971160626, 3.17478416773490358281526071205, 4.09687354714192189006385386327, 5.10788806922515596978156302282, 5.89328127291180924058359990219, 6.50380197718837386978751302448, 7.29159908640298022496151471022, 8.29148104704363001432758064589, 9.01600033463816344770707343940, 9.47147944725239268095319766949, 10.4903755277909849262824263722, 10.93121934983992978276474837804, 11.69599504736618367860185995245, 12.54205983012029316740474179144, 13.13635846789116511098866026650, 13.72579760652003828183860411227, 13.86421924592705290690285795855, 14.71654046645658003922029344561, 16.16143702158389055248815362970, 16.69611911445269437325044563170, 17.432776953234056601018623219520, 18.05741848629917036028895771547

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