Properties

Label 1-4008-4008.683-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.448 - 0.893i$
Analytic cond. $430.719$
Root an. cond. $430.719$
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  + (−0.898 − 0.438i)5-s + (0.982 + 0.188i)7-s + (−0.988 − 0.150i)11-s + (−0.942 + 0.334i)13-s + (0.489 − 0.872i)17-s + (0.351 + 0.936i)19-s + (−0.553 − 0.832i)23-s + (0.614 + 0.788i)25-s + (0.553 − 0.832i)29-s + (0.999 + 0.0378i)31-s + (−0.800 − 0.599i)35-s + (−0.843 + 0.537i)37-s + (0.997 + 0.0756i)41-s + (0.243 − 0.969i)43-s + (−0.965 − 0.261i)47-s + ⋯
L(s)  = 1  + (−0.898 − 0.438i)5-s + (0.982 + 0.188i)7-s + (−0.988 − 0.150i)11-s + (−0.942 + 0.334i)13-s + (0.489 − 0.872i)17-s + (0.351 + 0.936i)19-s + (−0.553 − 0.832i)23-s + (0.614 + 0.788i)25-s + (0.553 − 0.832i)29-s + (0.999 + 0.0378i)31-s + (−0.800 − 0.599i)35-s + (−0.843 + 0.537i)37-s + (0.997 + 0.0756i)41-s + (0.243 − 0.969i)43-s + (−0.965 − 0.261i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.170653847 - 0.7226446448i\)
\(L(\frac12)\) \(\approx\) \(1.170653847 - 0.7226446448i\)
\(L(1)\) \(\approx\) \(0.8915484586 - 0.08273921888i\)
\(L(1)\) \(\approx\) \(0.8915484586 - 0.08273921888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.898 - 0.438i)T \)
7 \( 1 + (0.982 + 0.188i)T \)
11 \( 1 + (-0.988 - 0.150i)T \)
13 \( 1 + (-0.942 + 0.334i)T \)
17 \( 1 + (0.489 - 0.872i)T \)
19 \( 1 + (0.351 + 0.936i)T \)
23 \( 1 + (-0.553 - 0.832i)T \)
29 \( 1 + (0.553 - 0.832i)T \)
31 \( 1 + (0.999 + 0.0378i)T \)
37 \( 1 + (-0.843 + 0.537i)T \)
41 \( 1 + (0.997 + 0.0756i)T \)
43 \( 1 + (0.243 - 0.969i)T \)
47 \( 1 + (-0.965 - 0.261i)T \)
53 \( 1 + (-0.726 + 0.686i)T \)
59 \( 1 + (0.489 + 0.872i)T \)
61 \( 1 + (-0.132 + 0.991i)T \)
67 \( 1 + (-0.898 + 0.438i)T \)
71 \( 1 + (0.0189 - 0.999i)T \)
73 \( 1 + (0.914 - 0.404i)T \)
79 \( 1 + (-0.316 - 0.948i)T \)
83 \( 1 + (0.776 + 0.629i)T \)
89 \( 1 + (-0.672 - 0.739i)T \)
97 \( 1 + (-0.999 + 0.0378i)T \)
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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

−18.30203360832627580741416219344, −17.72419436951317351628851819666, −17.3300708589158747988662593953, −16.19659108061825011579660867017, −15.66286682494257752125170420453, −15.04821238204524041192965191579, −14.427693478164708259350109174031, −13.83638426531235278736960918390, −12.74408974378700904137331234457, −12.32285397000321262999018852178, −11.4062528533775448434973289703, −10.96923383835835634210846387439, −10.25007269140448247778393667426, −9.548459502211855073741851984120, −8.29106663541600032308308009585, −8.01245624921306509652775394707, −7.375232802875519871530538230844, −6.669441860035297595478213398340, −5.49628141114962348420385626467, −4.90954022865592879495353591191, −4.248297023550486793234417126058, −3.26416119763809482337242593761, −2.62202699890613569299848758182, −1.64100296651105566164013517637, −0.58064296206735355708689684536, 0.33262050380743153181251840285, 1.19748613057418173131562526658, 2.28628732380103103968385888888, 2.95583393807097671351141277378, 4.04788511347901187845530385693, 4.75840421944941169052926505923, 5.174019642713017075296790551468, 6.11049114900981763481641651339, 7.27611703693373201320620628140, 7.77953307782119880395742550660, 8.24999945097055375248133594287, 9.02111609945312816876783004599, 10.014991233680336173535788295396, 10.553209369743460615536325585748, 11.53930938289373646959937705759, 12.05538234909761197066265612022, 12.37320765216911304021823557914, 13.53301186196762523453505267529, 14.1275929333992926223930652633, 14.8453246683409578472824814202, 15.488875224138691427110222000726, 16.17993970871597939830145400828, 16.69698540809986003065897024503, 17.55874667098551070731727979526, 18.23745970756702479976291953288

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