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Properties

Label 1-4008-4008.3005-r0-0-0
Degree $1$
Conductor $4008$
Sign $1$
Analytic cond. $18.6130$
Root an. cond. $18.6130$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 11-s + 13-s + 17-s − 19-s + 23-s + 25-s + 29-s + 31-s − 35-s + 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s + ⋯
L(s)  = 1  − 5-s + 7-s + 11-s + 13-s + 17-s − 19-s + 23-s + 25-s + 29-s + 31-s − 35-s + 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4008} (3005, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.177218963\)
\(L(\frac12)\) \(\approx\) \(2.177218963\)
\(L(1)\) \(\approx\) \(1.253585913\)
\(L(1)\) \(\approx\) \(1.253585913\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−18.62916211780674271232877545409, −17.75224908734863499612528175610, −17.09109783475013815944598423638, −16.49703115039160750270802317857, −15.671446373660192132845357470781, −15.0915990990548757721853353764, −14.38898116542481719626536778749, −13.973874281316793229341714876651, −12.79335334985979740108599887169, −12.31471143838209683621554302546, −11.35604887662353531831423803324, −11.22367894923485098568128701866, −10.36449504864993709134611214287, −9.31661529826275114582653593119, −8.55313062708104404368647791547, −8.122014764267576813518216122027, −7.39326340811516523150798412808, −6.52116188496488693360609761736, −5.866137888116141320478958458151, −4.65930038018110762276488039629, −4.362751478096098167039390348275, −3.48458011163534663217794598260, −2.69264655938549929476686282087, −1.38636222308207391208670532206, −0.933874570424816596986164704999, 0.933874570424816596986164704999, 1.38636222308207391208670532206, 2.69264655938549929476686282087, 3.48458011163534663217794598260, 4.362751478096098167039390348275, 4.65930038018110762276488039629, 5.866137888116141320478958458151, 6.52116188496488693360609761736, 7.39326340811516523150798412808, 8.122014764267576813518216122027, 8.55313062708104404368647791547, 9.31661529826275114582653593119, 10.36449504864993709134611214287, 11.22367894923485098568128701866, 11.35604887662353531831423803324, 12.31471143838209683621554302546, 12.79335334985979740108599887169, 13.973874281316793229341714876651, 14.38898116542481719626536778749, 15.0915990990548757721853353764, 15.671446373660192132845357470781, 16.49703115039160750270802317857, 17.09109783475013815944598423638, 17.75224908734863499612528175610, 18.62916211780674271232877545409

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