Properties

Label 1-4003-4003.356-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.504 - 0.863i$
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  + (−0.891 + 0.452i)2-s + (0.874 − 0.484i)3-s + (0.590 − 0.806i)4-s + (−0.817 − 0.576i)5-s + (−0.561 + 0.827i)6-s + (−0.922 − 0.386i)7-s + (−0.161 + 0.986i)8-s + (0.530 − 0.847i)9-s + (0.989 + 0.143i)10-s + (0.976 + 0.214i)11-s + (0.126 − 0.992i)12-s + (−0.947 − 0.319i)13-s + (0.997 − 0.0721i)14-s + (−0.994 − 0.108i)15-s + (−0.302 − 0.953i)16-s + (−0.725 + 0.687i)17-s + ⋯
L(s)  = 1  + (−0.891 + 0.452i)2-s + (0.874 − 0.484i)3-s + (0.590 − 0.806i)4-s + (−0.817 − 0.576i)5-s + (−0.561 + 0.827i)6-s + (−0.922 − 0.386i)7-s + (−0.161 + 0.986i)8-s + (0.530 − 0.847i)9-s + (0.989 + 0.143i)10-s + (0.976 + 0.214i)11-s + (0.126 − 0.992i)12-s + (−0.947 − 0.319i)13-s + (0.997 − 0.0721i)14-s + (−0.994 − 0.108i)15-s + (−0.302 − 0.953i)16-s + (−0.725 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.504 - 0.863i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.504 - 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014424674 - 0.5824054568i\)
\(L(\frac12)\) \(\approx\) \(1.014424674 - 0.5824054568i\)
\(L(1)\) \(\approx\) \(0.7952474630 - 0.1568908921i\)
\(L(1)\) \(\approx\) \(0.7952474630 - 0.1568908921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.891 + 0.452i)T \)
3 \( 1 + (0.874 - 0.484i)T \)
5 \( 1 + (-0.817 - 0.576i)T \)
7 \( 1 + (-0.922 - 0.386i)T \)
11 \( 1 + (0.976 + 0.214i)T \)
13 \( 1 + (-0.947 - 0.319i)T \)
17 \( 1 + (-0.725 + 0.687i)T \)
19 \( 1 + (0.958 + 0.284i)T \)
23 \( 1 + (-0.302 - 0.953i)T \)
29 \( 1 + (0.958 - 0.284i)T \)
31 \( 1 + (0.403 + 0.915i)T \)
37 \( 1 + (0.958 - 0.284i)T \)
41 \( 1 + (0.976 - 0.214i)T \)
43 \( 1 + (-0.0180 - 0.999i)T \)
47 \( 1 + (-0.302 + 0.953i)T \)
53 \( 1 + (-0.302 + 0.953i)T \)
59 \( 1 + (-0.773 - 0.633i)T \)
61 \( 1 + (-0.436 + 0.899i)T \)
67 \( 1 + (0.997 + 0.0721i)T \)
71 \( 1 + (0.874 - 0.484i)T \)
73 \( 1 + (0.647 + 0.762i)T \)
79 \( 1 + (-0.436 - 0.899i)T \)
83 \( 1 + (-0.891 + 0.452i)T \)
89 \( 1 + (0.997 - 0.0721i)T \)
97 \( 1 + (0.997 + 0.0721i)T \)
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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

−18.69389928704862401058081867093, −18.2402089669828501335127153357, −17.223427087626137601863780778632, −16.41775897215809360388515461211, −15.882802506623811604636233911857, −15.43812885444825091260125691166, −14.64401270844829928858192347268, −13.87537234159700947025612379389, −13.073567675598369740226287555097, −12.17584319493240934268875156697, −11.56472738873758782049178467456, −11.0847238318071230557085534984, −9.89910463383298233203356989622, −9.64273282486967332897397300288, −9.090062388121695910841500502480, −8.232342813653231615759745711183, −7.55040351834428165889036615153, −6.938348405090186728325671387334, −6.280930436465377262904734242739, −4.77629186043682095345026144009, −4.001645332780636850110342142546, −3.2660254278951740245399814673, −2.79278292852613166065826012265, −2.07375677809437932261790580871, −0.75151535199685559743759888609, 0.600312322918566119604362819743, 1.23941404725392103168477148853, 2.33155355803283554442303185235, 3.11322410372356032623470597696, 4.05142126699300242272242210525, 4.746280617671245613163106012688, 6.09221475927130704210850033323, 6.636598826041326960659243359857, 7.37450965344660378735598655719, 7.83716969820207878294347624327, 8.65740285647345638898832580392, 9.180122240423390953616249129107, 9.79891104615656977688468490095, 10.50176457955189944464936390501, 11.55511777226108659992406094548, 12.43839171608470046453378902233, 12.5900343693834393500420257732, 13.867628172053252530529791435947, 14.32606459390931034561268302840, 15.13914150099718040337435097380, 15.72410485231392131376090005981, 16.2700538379885081265025749716, 17.10661667911579470425371243578, 17.578357136368544195325285345, 18.51969379757304393126791317105

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