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Properties

Label 1-4004-4004.1367-r0-0-0
Degree $1$
Conductor $4004$
Sign $0.419 - 0.907i$
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  + (−0.913 − 0.406i)3-s + (−0.743 − 0.669i)5-s + (0.669 + 0.743i)9-s + (0.406 + 0.913i)15-s + (−0.309 − 0.951i)17-s + (0.994 − 0.104i)19-s + 23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.951i)27-s + (0.913 − 0.406i)29-s + (−0.743 + 0.669i)31-s + (0.587 − 0.809i)37-s + (−0.994 + 0.104i)41-s + (−0.5 − 0.866i)43-s − i·45-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)3-s + (−0.743 − 0.669i)5-s + (0.669 + 0.743i)9-s + (0.406 + 0.913i)15-s + (−0.309 − 0.951i)17-s + (0.994 − 0.104i)19-s + 23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.951i)27-s + (0.913 − 0.406i)29-s + (−0.743 + 0.669i)31-s + (0.587 − 0.809i)37-s + (−0.994 + 0.104i)41-s + (−0.5 − 0.866i)43-s − i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.419 - 0.907i$
Analytic conductor: \(18.5944\)
Root analytic conductor: \(18.5944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (1367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (0:\ ),\ 0.419 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8636083743 - 0.5523927330i\)
\(L(\frac12)\) \(\approx\) \(0.8636083743 - 0.5523927330i\)
\(L(1)\) \(\approx\) \(0.7234501284 - 0.2065484629i\)
\(L(1)\) \(\approx\) \(0.7234501284 - 0.2065484629i\)

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 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 + T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.994 + 0.104i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.743 + 0.669i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−18.361924889732246876522051054992, −18.104272471293808599230555345419, −17.09796086186509049906194118941, −16.64632201235676832802849469292, −15.82199725049844091232940856527, −15.28638855434547352053164274228, −14.786353735437012557728466203345, −13.90905296495781367938066811596, −12.91880025447602175353190945409, −12.35112234996308808666550147455, −11.482436171240928046517395273959, −11.19611308598031694730271150759, −10.41208736377957056930387296788, −9.83682770418803925800125552801, −8.9380038364467516427318570410, −8.04126707448686766128645673649, −7.2884008662760432489308118121, −6.58643006807761170027644779339, −6.01366067050412399024891346002, −5.02264490956463521404127382636, −4.45493353127320460088280759868, −3.54114923254783024173423913405, −3.02970022502117212496052632739, −1.70209470608128115978569595447, −0.68128315815770165634735435140, 0.584046207634087987112909784031, 1.17886896077032908185499258082, 2.30122804639511347756642839229, 3.34316731740666719596162966224, 4.24036015199962518168880677439, 5.13834057656143793536531210424, 5.29198464936891101296821122376, 6.49824702606043523400439334612, 7.194252922403454206387220052592, 7.64982281300520658377852575919, 8.62334737665644902563031522231, 9.26522270541978574135975062491, 10.180207946904505980794309729198, 11.00539654010327986182963226619, 11.58509423158463036393785381565, 12.1201945681159296409032291968, 12.71330863242882262006993164720, 13.49659838482853647948279067195, 14.04393628894270473269544125073, 15.36255008139298530544427786262, 15.6175625240579602151644745754, 16.56558517317326050400042420579, 16.78038568113286445388084028992, 17.70815553844721570909338529924, 18.37194417790189280109342057809

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