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Properties

Label 1-4003-4003.127-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.941 - 0.338i$
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  + (0.700 + 0.713i)2-s + (−0.817 − 0.576i)3-s + (−0.0180 + 0.999i)4-s + (0.997 − 0.0721i)5-s + (−0.161 − 0.986i)6-s + (0.403 − 0.915i)7-s + (−0.725 + 0.687i)8-s + (0.336 + 0.941i)9-s + (0.750 + 0.661i)10-s + (0.468 + 0.883i)11-s + (0.590 − 0.806i)12-s + (0.0541 − 0.998i)13-s + (0.935 − 0.353i)14-s + (−0.856 − 0.515i)15-s + (−0.999 − 0.0361i)16-s + (0.796 − 0.605i)17-s + ⋯
L(s)  = 1  + (0.700 + 0.713i)2-s + (−0.817 − 0.576i)3-s + (−0.0180 + 0.999i)4-s + (0.997 − 0.0721i)5-s + (−0.161 − 0.986i)6-s + (0.403 − 0.915i)7-s + (−0.725 + 0.687i)8-s + (0.336 + 0.941i)9-s + (0.750 + 0.661i)10-s + (0.468 + 0.883i)11-s + (0.590 − 0.806i)12-s + (0.0541 − 0.998i)13-s + (0.935 − 0.353i)14-s + (−0.856 − 0.515i)15-s + (−0.999 − 0.0361i)16-s + (0.796 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.941 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.440422610 - 0.4252464282i\)
\(L(\frac12)\) \(\approx\) \(2.440422610 - 0.4252464282i\)
\(L(1)\) \(\approx\) \(1.485439582 + 0.1630613290i\)
\(L(1)\) \(\approx\) \(1.485439582 + 0.1630613290i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.700 + 0.713i)T \)
3 \( 1 + (-0.817 - 0.576i)T \)
5 \( 1 + (0.997 - 0.0721i)T \)
7 \( 1 + (0.403 - 0.915i)T \)
11 \( 1 + (0.468 + 0.883i)T \)
13 \( 1 + (0.0541 - 0.998i)T \)
17 \( 1 + (0.796 - 0.605i)T \)
19 \( 1 + (0.126 + 0.992i)T \)
23 \( 1 + (-0.999 - 0.0361i)T \)
29 \( 1 + (0.126 - 0.992i)T \)
31 \( 1 + (0.874 - 0.484i)T \)
37 \( 1 + (0.126 - 0.992i)T \)
41 \( 1 + (0.468 - 0.883i)T \)
43 \( 1 + (-0.0901 - 0.995i)T \)
47 \( 1 + (-0.999 + 0.0361i)T \)
53 \( 1 + (-0.999 + 0.0361i)T \)
59 \( 1 + (0.958 + 0.284i)T \)
61 \( 1 + (-0.773 - 0.633i)T \)
67 \( 1 + (0.935 + 0.353i)T \)
71 \( 1 + (-0.817 - 0.576i)T \)
73 \( 1 + (-0.370 - 0.928i)T \)
79 \( 1 + (-0.773 + 0.633i)T \)
83 \( 1 + (0.700 + 0.713i)T \)
89 \( 1 + (0.935 - 0.353i)T \)
97 \( 1 + (0.935 + 0.353i)T \)
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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

−18.55743482306996021218225889369, −17.8690139698466631830419922567, −17.292463166645832289979133275143, −16.26623858603557832703310354732, −15.95193756102984226000054488557, −14.718101642914269743895821292261, −14.57696054863281899762532378100, −13.73501259864871560312198112515, −12.95512326695752988857778354697, −12.201246373418235332198887414865, −11.56064180381262614829253968000, −11.185200670148214854295516625518, −10.316254859391672232133177092432, −9.68413809168271834531128886533, −9.13378920414203420827118617548, −8.39059462778013154770151274492, −6.70670606519627676537253003411, −6.25201153807539902149584405673, −5.74598353803838738609364755300, −4.95143374100618698127606846375, −4.48730254700429678656333497796, −3.37442923164904061589894638905, −2.76723822765371081141322377667, −1.6278264667931376432410387931, −1.16458114227693651083871147665, 0.62622712517753904328319611487, 1.68459806755180030527351816826, 2.40488106924094600938034677867, 3.61766220581150805154777710584, 4.44337158986954532324365667128, 5.13279486792115192836619640047, 5.81153783599887415723920405136, 6.29525497225808048143680759088, 7.166475347716533364344292859096, 7.69941950302580881902970592389, 8.27194229666882555870122492168, 9.653136988423502882971927902771, 10.1281021847002526175979600179, 10.93958825968066418509196089182, 11.94919320209333307805441020103, 12.32415355048591964696735676668, 13.0813678510199750479921032246, 13.73177917488229759492183742210, 14.185039438693364960243714943124, 14.85732408226761612989321283417, 15.921776925800843503710226852361, 16.47918271926227294900205367917, 17.25628984919227432378871219173, 17.51827293707449266300636749870, 18.01035807757829689990350941837

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