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Properties

Label 1-4008-4008.77-r1-0-0
Degree $1$
Conductor $4008$
Sign $-0.898 - 0.439i$
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  + (0.726 − 0.686i)5-s + (0.206 − 0.978i)7-s + (−0.700 − 0.713i)11-s + (0.843 + 0.537i)13-s + (0.752 + 0.658i)17-s + (−0.997 − 0.0756i)19-s + (−0.898 − 0.438i)23-s + (0.0567 − 0.998i)25-s + (0.898 − 0.438i)29-s + (0.553 + 0.832i)31-s + (−0.521 − 0.853i)35-s + (0.584 + 0.811i)37-s + (0.387 − 0.922i)41-s + (0.993 + 0.113i)43-s + (−0.822 − 0.569i)47-s + ⋯
L(s)  = 1  + (0.726 − 0.686i)5-s + (0.206 − 0.978i)7-s + (−0.700 − 0.713i)11-s + (0.843 + 0.537i)13-s + (0.752 + 0.658i)17-s + (−0.997 − 0.0756i)19-s + (−0.898 − 0.438i)23-s + (0.0567 − 0.998i)25-s + (0.898 − 0.438i)29-s + (0.553 + 0.832i)31-s + (−0.521 − 0.853i)35-s + (0.584 + 0.811i)37-s + (0.387 − 0.922i)41-s + (0.993 + 0.113i)43-s + (−0.822 − 0.569i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ -0.898 - 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4747077991 - 2.050474448i\)
\(L(\frac12)\) \(\approx\) \(0.4747077991 - 2.050474448i\)
\(L(1)\) \(\approx\) \(1.125860806 - 0.4262141068i\)
\(L(1)\) \(\approx\) \(1.125860806 - 0.4262141068i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.726 - 0.686i)T \)
7 \( 1 + (0.206 - 0.978i)T \)
11 \( 1 + (-0.700 - 0.713i)T \)
13 \( 1 + (0.843 + 0.537i)T \)
17 \( 1 + (0.752 + 0.658i)T \)
19 \( 1 + (-0.997 - 0.0756i)T \)
23 \( 1 + (-0.898 - 0.438i)T \)
29 \( 1 + (0.898 - 0.438i)T \)
31 \( 1 + (0.553 + 0.832i)T \)
37 \( 1 + (0.584 + 0.811i)T \)
41 \( 1 + (0.387 - 0.922i)T \)
43 \( 1 + (0.993 + 0.113i)T \)
47 \( 1 + (-0.822 - 0.569i)T \)
53 \( 1 + (0.672 - 0.739i)T \)
59 \( 1 + (-0.752 + 0.658i)T \)
61 \( 1 + (-0.954 - 0.298i)T \)
67 \( 1 + (-0.726 - 0.686i)T \)
71 \( 1 + (0.881 + 0.472i)T \)
73 \( 1 + (-0.169 + 0.985i)T \)
79 \( 1 + (0.489 + 0.872i)T \)
83 \( 1 + (0.421 - 0.906i)T \)
89 \( 1 + (0.942 - 0.334i)T \)
97 \( 1 + (0.553 - 0.832i)T \)
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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

−18.41996565576451872248557418622, −17.97406102219223003999212522485, −17.57485208560736783549865803753, −16.491293997648720326324002102201, −15.77185313091588448714256025715, −15.166136814515776169097155848497, −14.59118202765743696242318058641, −13.86275289842760542707718615604, −13.13075335728032062622559966241, −12.47719079080976504471416269725, −11.76930697109774452454250281376, −10.8797960405076379273428601207, −10.375332157798843566428208865106, −9.59611403488935733150414252234, −9.02689304309118093264419274750, −7.95114044958836391112082554758, −7.62572053216522073236578401603, −6.33098303298779635535786308692, −6.049157453336631183574724509593, −5.25943358117371198880282167307, −4.46381199002620548727508325619, −3.33111634685151534554708096657, −2.590201470079398553757603122376, −2.07747326450239470210129568659, −1.06114205499456586414596987716, 0.31657114072462827561796556924, 1.0934735957633384481930087640, 1.82029675483069091544251954367, 2.79405014288127404473963164712, 3.8417372697004070071542904976, 4.42098635511906626091325111444, 5.23051242647286177056928938320, 6.17561697399256338013892441181, 6.45692719213286323473159864163, 7.7107119922817791619662229364, 8.34478378819039852030250021784, 8.78220764791925779737976009943, 9.89669144742141774655921124679, 10.39852817992253342523006683764, 10.92276577931752074605456806446, 11.91424465125507456244850908837, 12.66852178973245511712143047454, 13.33129576081249779057811114078, 13.90504449513709950345743326984, 14.309046934580094262466493757373, 15.41207461556859924448134681550, 16.24210953143457668145812036123, 16.60583801099509792852780713823, 17.29238477230107541798163998172, 17.91840084541388274438972786131

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