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Properties

Label 1-4004-4004.1399-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.00182 - 0.999i$
Analytic cond. $18.5944$
Root an. cond. $18.5944$
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  + (0.309 − 0.951i)3-s + (0.587 + 0.809i)5-s + (−0.809 − 0.587i)9-s + (0.951 − 0.309i)15-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s + 23-s + (−0.309 + 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.587 − 0.809i)31-s + (0.951 − 0.309i)37-s + (0.951 + 0.309i)41-s − 43-s − i·45-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (0.587 + 0.809i)5-s + (−0.809 − 0.587i)9-s + (0.951 − 0.309i)15-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s + 23-s + (−0.309 + 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.309 − 0.951i)29-s + (0.587 − 0.809i)31-s + (0.951 − 0.309i)37-s + (0.951 + 0.309i)41-s − 43-s − i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00182 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00182 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.00182 - 0.999i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.00182 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389015001 - 1.391556077i\)
\(L(\frac12)\) \(\approx\) \(1.389015001 - 1.391556077i\)
\(L(1)\) \(\approx\) \(1.203814485 - 0.3851712706i\)
\(L(1)\) \(\approx\) \(1.203814485 - 0.3851712706i\)

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 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−18.7361560872983773034547567521, −17.78982496807019131977760712887, −16.97881110419417383659111845682, −16.74348927633462478144969264533, −16.009218312903040314448611182756, −15.25428706891879853684535751541, −14.51906836431517762933037979313, −14.11017749469546351364717791035, −13.04179130670072332981716194388, −12.723402101669125003789825863450, −11.74523330818021625507012556385, −10.877058364541605455556903576357, −10.27657317091028416405488931094, −9.65353424431933057524243756484, −8.941368984363648789558730433803, −8.430713029833732438412073943317, −7.72166633791655073360087016686, −6.50456667138497492061892202082, −5.81286371324329256200897555503, −5.02864581051334507648642852407, −4.551170883654306906773601643113, −3.63781491541290055771315361965, −2.89501017555259983276329632636, −1.93028136136830407885812431506, −1.05989851212102082395232634659, 0.55318885536140881263439793419, 1.58989072440941993863178514675, 2.42060557229198555408984964001, 2.91002577719444231136878078784, 3.788799356013241764023427716251, 4.93749080302913960751996255930, 5.84868688072382563770377637932, 6.41903802913447367121365746723, 7.032711414873694158288752526339, 7.776209512098686909882898449258, 8.38455499817187404178615703305, 9.46756753986938985063910650209, 9.76733317768022059493150586943, 10.95989234047100577177138965075, 11.34823510090357637327180262909, 12.21253722733548683904769022686, 13.072774019542072398943581176876, 13.428346015596470440238619714771, 14.19881673233224008990616445806, 14.85621033638884095981296727742, 15.22501241122676280118574137375, 16.51459001088316440862436653753, 17.09337856639007625842509970844, 17.787904999370890128500477915318, 18.31892298292383546377123804357

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