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Properties

Label 1-4004-4004.1879-r1-0-0
Degree $1$
Conductor $4004$
Sign $-0.954 - 0.298i$
Analytic cond. $430.289$
Root an. cond. $430.289$
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 + (−0.994 + 0.104i)5-s + (−0.104 + 0.994i)9-s + (−0.743 − 0.669i)15-s + (−0.809 + 0.587i)17-s + (−0.207 − 0.978i)19-s + 23-s + (0.978 − 0.207i)25-s + (−0.809 + 0.587i)27-s + (0.669 − 0.743i)29-s + (−0.994 − 0.104i)31-s + (0.951 − 0.309i)37-s + (0.207 + 0.978i)41-s + (−0.5 + 0.866i)43-s − i·45-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)3-s + (−0.994 + 0.104i)5-s + (−0.104 + 0.994i)9-s + (−0.743 − 0.669i)15-s + (−0.809 + 0.587i)17-s + (−0.207 − 0.978i)19-s + 23-s + (0.978 − 0.207i)25-s + (−0.809 + 0.587i)27-s + (0.669 − 0.743i)29-s + (−0.994 − 0.104i)31-s + (0.951 − 0.309i)37-s + (0.207 + 0.978i)41-s + (−0.5 + 0.866i)43-s − i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.954 - 0.298i$
Analytic conductor: \(430.289\)
Root analytic conductor: \(430.289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (1:\ ),\ -0.954 - 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1302432103 + 0.8530470264i\)
\(L(\frac12)\) \(\approx\) \(-0.1302432103 + 0.8530470264i\)
\(L(1)\) \(\approx\) \(0.9420173051 + 0.3529057216i\)
\(L(1)\) \(\approx\) \(0.9420173051 + 0.3529057216i\)

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 \( 1 + (-0.994 + 0.104i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 + T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.994 - 0.104i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (0.743 + 0.669i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−18.22359857805782309166958416424, −17.3163096214984523030958736521, −16.52661710791728782950371392100, −15.81321155490516110090468865145, −15.126902128598045650698764029616, −14.55793298419017009047896053088, −13.8852373198397470184828995165, −13.02469395216602427348031894031, −12.549377483375823126749896069363, −11.8657859123167243694315305177, −11.19535769834611768649921196630, −10.41783565301687176549915818072, −9.32672439978243780654138078729, −8.7526928445277609833818403181, −8.20525654741547999460979290633, −7.30405989914476034282809437258, −7.03231829465678783731692154928, −6.0862090396830508486817616362, −5.07561380572167948181842855106, −4.203505484039170509408174862759, −3.48409732097260564763383603052, −2.79772736346616661580512805387, −1.87921839745230197494433851223, −0.952750815307441581161122765294, −0.14158053880734206251981455057, 0.96096773041096410501222505545, 2.26428494464452931879259153203, 2.89803801651844633201048480061, 3.695282536052584356448845863212, 4.437030354255480242362614027295, 4.83246385391585493147357624988, 5.98212351472934476560720698578, 6.949751906076959469587472573337, 7.58017415501677977867274063850, 8.3995620378418370596391924537, 8.84673644750853857373674338580, 9.60414658984861174411571520636, 10.46190351295373742435402192706, 11.20689818508407359892737407921, 11.45416802055974485853860695581, 12.76390491411611831586352511653, 13.12059641325019539917371567665, 14.078911589737083034906931143308, 14.82259439928715572876743378782, 15.29677910818423320831576488690, 15.74726642507706028068187836198, 16.55148067986896491339281780748, 17.13245969819406252832679005095, 18.09484476847821302290726077913, 18.88651724342456973781418789808

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